Is it prescribed for a person who sneezes, when he praises Allah, to say “Yarhamuk Allah (May Allah have mercy on you)” to himself?

The Sunnah is to say to the one who sneezes, “Yarhamuk
Allah (may Allah have mercy on you)” if he sneezes and praises Allah (by
saying “al-hamdu lillah (praise be to Allah”). If he does not praise
Allah, then one should not say “Yarhamuk Allah” to him. 

Al-Bukhaari (6221) and Muslim (2991) narrated that Anas ibn
Maalik (may Allah be pleased with him) said: Two men sneezed in the presence
of the Prophet (blessings and peace of Allah be upon him) and he said
“Yarhamuk Allah” to one of them and not to the other. He was asked about
that and he said: “This one praised Allah and that one did not praise
Allah.” 

And Muslim (2992) narrated that Abu Moosa (may Allah be
pleased with him) said: I heard the Messenger of Allah (blessings and peace
of Allah be upon him) say: “When one of you sneezes and praises Allaah, then
say Yarhamuk Allaah to him, but if he does not praise Allaah, then do
not say it to him.” End quote.  

Al-Nawawi (may Allah have mercy on him) said: 

This is a clear instruction to say “Yarhamuk Allah” if the
one who sneezes praises Allah, and clearly forbids doing so if he does not
praise Him. End quote. 

Secondly: 

if a non-Muslim sneezes and praises Allah, one should not say
to him, “Yarhamuk Allah (may Allah have mercy on you)”; rather one should
say to him: “Yahdeekum Allahu wa yuslih baalakum (may Allah guide you and
set your affairs straight),” because of the reports narrated by Abu Dawood
(5038), and by al-Tirmidhi (2739) who classed it as saheeh, from Abu Moosa
(may Allah be pleased with him), who said: The Jews used to sneeze in the
presence of the Prophet (blessings and peace of Allah be upon him) hoping
that he would say to them, “May Allaah have mercy on you,” but he used to
say: “May Allaah guide you and set your affairs straight.”.

Classed as saheeh by al-Albaani in Saheeh Abi Dawood. 

Al-Haafiz (may Allah have mercy on him) said in al-Fath
(10/604): 

The hadeeth of Abu Moosa indicates that they are included in
the general command to pray for the one who sneezes, but there is a special
prayer for them, which is the prayer that they be guided and their affairs
set straight. There is nothing wrong with that, and it is different from the
prayer for Muslims, for they are the ones for whom mercy should be sought,
unlike the kuffaar. End quote. 

Thirdly: 

If a Muslim sneezes and praises Allah, it is the same whether
someone prays for mercy for him or not, whether he is alone or is in a
gathering of people. It is not prescribed for him to pray for mercy for
himself (in this situation); rather he should praise Allah only (by saying
“al-hamdu Lillah”). 

Praying for mercy for oneself after saying “al-hamdu lillah”
is not narrated in the Sunnah and it is not narrated from any of the
Sahaabah as far as we know. 

Al-Bukhaari and Muslim (1718) narrated that ‘Aa’ishah (may
Allah be pleased with her) said: The Messenger of Allah (blessings and peace
of Allah be upon him) said: “Whoever introduces anything into this matter of
ours that is not part of it will have it rejected.” 

If this is what was prescribed, the Prophet (blessings and
peace of Allah be upon him) would have stated that, especially when there is a need to know the ruling concerning the matter. 

And Allah knows best.