Does seeing the new moon during the day count with regard to verifying the beginning or end of the month?

Firstly: 

Allah has made the moon the sign of the night, which is the
time when it dominates and appears. He, may He be exalted, says
(interpretation of the meaning):

“And We have appointed the night and the day as two Ayat
(signs etc.). Then, We have made dark the sign of the night while We have
made the sign of day illuminating, that you may seek bounty from your Lord,
and that you may know the number of the years and the reckoning. And We have
explained everything (in detail) with full explanation”

[al-Isra’ 17:12]. 

Ibn Katheer said:

Allah has given the night a sign, i.e., a feature by which it
is known, which is darkness and the appearance of the moon. And the day also
has a sign, which is light and the appearance of the brightly shining sun.
And He made a difference between the light of the moon and the light of the
sun so that the one may be distinguished from the other, as He, may He be
exalted, says (interpretation of the meaning):

“It is He Who made the sun a shining thing and the moon as a
light and measured out its (their) stages, that you might know the number of
years and the reckoning. Allah did not create this but in truth”

[Yoonus 10:5]. 

End quote from Tafseer Ibn Katheer (5/50) 

Therefore, all rulings having to do with the moon can only be
proven by seeing it at night, not during the day. 

Abu’l-Hasanaat al-Laknawi said:

This indicates that the moon is indeed the sign of the night,
and not the sign of the day, so sighting it during the day is of no
significance. It also indicates that the new moons are signs to mark fixed
periods of time for mankind and for the pilgrimage, fasting and so on, and
to know the number of years and the reckoning and the like. But this is only
when it rises during the night, and not otherwise.

End quote from al-Falak al-Dawwaar fi Ru’yat al-Hilaal
fi’n-Nahaar (p. 18) 

Therefore the fuqaha’ clearly stated that the lunar eclipse
prayer is not to be offered if the moon appears eclipsed during the day,
because it is not dominant at that time. an-Nawawi (may Allah have mercy on
him) said: If the lunar eclipse begins after sunrise, then (the eclipse
prayer) should not be offered, and there is no difference of opinion
concerning that (in our madhhab). 

End quote from al-Majmoo‘ Sharh al-Muhadhdhab (5/54)

Secondly: 

The majority of scholars – and this is the established view
among the four madhhabs – are of the view that sighting the new moon during
the day does not lead to any rulings. So if a fasting person sees it during
the thirtieth day of Ramadan, he should continue with his fast and not break
it, and if one who is not fasting on the thirtieth day of Sha‘baan sees it,
he should not stop eating and drinking, or make up that day. Sighting the
new moon during the day is of no significance; rather what matters is
sighting the new moon after sunset only.

In Musannaf Ibn Abi Shaybah (3/67), it is narrated
with a saheeh isnaad from Abu Waa’il that he said: A letter came to us from
‘Umar when we were in Khaaniqeen, saying: Some new moons are bigger than
others, so if you see the new moon during the day, do not break the fast,
unless two Muslim men testify that they had seen it the night before. End
quote. 

Al-Bayhaqi narrated with a saheeh isnaad from Saalim ibn
‘Abdullah ibn ‘Umar that some people saw the new moon of (Eid) al-Fitr
during the day, but ‘Abdullah ibn ‘Umar (may Allah be pleased with him)
continued his fast until the night, and he said: No, not until it is seen
when it is usually seen, at night. End quote from Sunan al-Bayhaqi
(2/435). 

In al-Fataawa al-Hindiyyah (1/197) it says:

If they see the new moon before or after the meridian, they
should not fast or break the fast on that basis. End quote. 

Is-haaq ash-Shiraazi said:

Fasting and breaking the fast are not connected to anything
except what we see after sunset.

End quote from al-Muhadhdhab (3/33) 

Shams ad-Deen ar-Ramli said:

If it is seen on the day of the twenty-ninth, but it was not
seen at night, then there is no scholar who says that seeing it during the
day is of any significance. Therefore it is clear that sighting it during
the day has no consequences.

End quote from Fataawa ar-Ramli (2/78) 

In Kashshaaf al-Qinaa‘ (2/303) it says: Sighting of
the new moon during the day has no consequences; it only counts if it is
sighted after sunset. End quote. 

Al-Laknawi said: The imams of the four madhhabs clearly
stated that the correct view is that sighting of the new moon during the day
does not count; rather what counts is sighting it at night.

End quote from al-Falak ad-Dawwaar (p. 19) 

The hadiths that speak of beginning and ending the fast
according to the sighting of the new moon only refer to sighting it at
night, not during the day. Siddeeq Hasan Khan said: The sighting of the new
moon that the Lawgiver referred to in the words “Fast when you see it” is
sighting it by night, not sighting it during the day, for that does not
count, regardless of whether it is before or after the meridian. Whoever
claims anything contrary to that is far away from knowledge of the
objectives of sharee‘ah.

End quote from ar-Rawdah an-Nadiyyah (2/11) 

Abu’l-Hasanaat al-Laknawi said:

Some of them claimed that any sighting of the new moon means
that it is obligatory to break the fast, because of the hadith: “Stop
fasting when you see it,” without differentiating between night and day.
They forget that what is referred to in the hadiths is the regular sighting,
which is sighting at night, not during the day.

End quote from al-Falak ad-Dawwaar (p. 9) 

Shaykh Ibn ‘Uthaymeen said:

The beginning of the month can only be when the new moon is
seen after sunset, following it.

End quote from Majmoo‘ Fataawa wa Rasaa’il al-‘Uthaymeen
(16/301) 

Thirdly: 

According to many of the fuqaha’, if the new moon is sighted
during the day, it belongs to the coming night, not the previous night, and
they did not mean that there are any consequences to seeing it during the
day, because what they are referring to in these words is the sighting of
the new moon that happens on the thirtieth day of Sha‘baan or Ramadaan. In
that case they ruled that it belongs to the coming night, because the month
has been completed when it reached thirty days. So it comes under the
heading of telling of a real situation, not determining the ruling on the
basis of sighting the new moon during the day. And it is a refutation of
those who say that it belongs to the previous night, such as al-Qaadi Abu
Yoosuf (may Allah have mercy on him). Imam an-Nawawi (may Allah have mercy
on him) said: If they sight the new moon during the day, then it belongs to
the coming night, whether they see it before or after the meridian. This is
our view, and there is no difference of opinion concerning it. This was also
the view of Abu Haneefah, Maalik and Muhammad. End quote from al-Majmoo‘
(6/279). Something similar was also stated in al-Mughni by Ibn
Qudaamah (3/173). 

Al-Qalyoobi said: Sighting of the new moon during the day
does not have any consequences; i.e., it does not belong to the previous
night, so that the fast may be broken, and it does not belong to the coming
night so that it may be proven that Ramadan has begun, for example. Whoever
says that it belongs to the coming night, that is correct if it is sighted
on the thirtieth day of the month, but it makes no difference, because the
month has been completed, which is different from the situation on the
twenty-ninth day, which does not mean that there is no need for it to be
sighted after sunset on the coming night, as some of them think. 

End quote from Haashiyat ‘ala Kanz ar-Raaghibeen
(2/65) 

Ibn ‘Aabideen said: The fact that it is sighted during the
day does not definitely mean that it belongs to the coming night, because
sighting of the new moon during the day does not count in their view [namely
Abu Haneefah and his companion Muhammad ibn al-Hasan]. Rather it is proven
by the completion of the number of days (of the month), because the
difference of opinion – according to what is clearly stated in al-Badaa’i‘
and al-Fath – has to do only with sighting of the new moon on the day
of doubt, which is the thirtieth day of Sha‘baan or Ramadan. If the Friday
mentioned was the thirtieth day of the month, and the new moon was sighted
during the day, then according to Abu Yoosuf that day is the first of the
month, but according to them [Abu Haneefah and Muhammad ibn al-Hasan] this
sighting is of no significance, and the first day of the month is Saturday,
whether the new moon is sighted during the day or not, because the month
cannot be more than thirty days. So this sighting does not affect anything.
In that case, their saying that this sighting belongs to the coming night,
in their view, is simply stating the obvious reality and is a statement on
their part that they differ with the view that it belongs to the previous
night. In that case, there is no contradiction between their statement that
it belongs to the coming night and their statement that sighting it during
the day is of no significance. Rather the difference of opinion has to do
with sighting it on the day of doubt, which is the thirtieth day of the
month, because if it is sighted on the twenty-ninth, no one says that it
belongs to the previous night, because that would imply that the month has
twenty-eight days, as was stated by some of the scholars.

End quote from Haashiyat Ibn ‘Aabideen (2/392) 

Shaykh Ibn ‘Uthaymeen said, commenting on the words of al-Hajaawi,
“If it is sighted during the day, then it belongs to the coming night”: The
pronoun refers to the new moon, and the author did not mean that it belongs
to the coming night (in the sense that the next day is the beginning of the
next month, based on sighting it during the day); rather he meant to negate
the view of those who say that it belongs to the previous night, because
some of the scholars said: if the new moon is sighted during the day before
sunset on that day, then it belongs to the previous night, so the people
must stop eating and drinking. Some of the scholars differentiated between
whether it is sighted before or after the meridian, but the correct view is
that it does not belong to the previous night.

End quote from ash-Sharh al-Mumti‘ (6/307)