How is the fast of Dawood (peace be upon him) observed, and how can that be reconciled with the prohibition on fasting on Fridays?

Firstly: It is narrated in al-Saheehayn from
‘Abd-Allaah ibn ‘Amr (may Allaah be pleased with him) that the Prophet
(peace and blessings of Allaah be upon him) said: “The best fasting is the
fast of Dawood: he used to fast one day and not the next.” This statement
does not contradict the fact that it is not allowed to fast on Fridays,
because the prohibition on fasting on Fridays applies to one who singles out
a Friday rather than any other day. But the one who observes the fast of
Dawood – fasting alternate days – is not singling out a Friday for fasting. 

Shaykh Muhammad ibn Saalih al-‘Uthaymeen (may Allaah have
mercy on him) said:  

The hadeeth of ‘Abd-Allaah ibn ‘Amr (may Allaah be pleased
with him) indicates that if fasting on a Friday or Saturday coincides with a
day one usually fasts, without intending to single that day out for fasting,
then there is nothing wrong with it, because if he fasts alternate days,
then that will coincide with some Fridays and Saturdays. Thus it becomes
clear that fasting on these days is not haraam. Otherwise the Prophet
(peace and blessings of Allaah be upon him) would have said: Fast one day
and not the next, so long as that (fasting) does not coincide with a Friday
or Saturday. 

Al-Sharh al-Mumti’, 6/476. 

Secondly: With regard to
your question about the ruling on singling out Friday for fasting in the law
of Dawood (peace be upon him): 

We do not know anything about the practical rulings of the
law of Dawood (peace be upon him) with regard to whether it was forbidden to
fast on Fridays or other days. What we do know is that every Prophet has a
law and a way, but their belief was one and the same, although their laws
differed. 

Allaah says (interpretation of the meaning):

“To each among you, We have
prescribed a law and a clear way”

[al-Maa’idah 5:48]

It was narrated that Abu Hurayrah (may Allaah be pleased with
him) said: I heard the Messenger of Allaah (peace and blessings of
Allaah be upon him) say: “The Prophets are paternal brothers; their mothers
are different, but their religion is the same.” Narrated by al-Bukhaari,
3259; Muslim, 2365. 

What this hadeeth
means is that the religion of the Prophets is one, which is Tawheed or
belief in the Oneness of Allaah and that He alone is entitled to worship,
even though their laws may differ. This is like brothers who have the same
father but different mothers (which is what is meant by paternal brothers).