Firstly: 

Muslim (1162) narrated from Abu Qataadah al-Ansaari (may Allah be pleased with him) that the Messenger of Allah (blessings and peace of Allah be upon him) was asked about fasting on Mondays and he said: “On (that day) I was born and on it Revelation came down to me.” 

Al-Tirmidhi (747) narrated, in a hadeeth that he classed as hasan, from Abu Hurayrah (may Allah be pleased with him) that the Messenger of Allah (blessings and peace of Allah be upon him) said: “Deeds are presented (to Allah) on Monday and Thursday and I like my deeds to be presented when I am fasting.”

Classed as saheeh by al-Albaani in Saheeh al-Tirmidhi. 

From the saheeh hadeeths quoted above, it is clear that just as the Prophet (blessings and peace of Allah be upon him) fasted on Mondays out of gratitude for the blessing of his birth on this day, he also fasted it because of its virtue, because the Revelation came down to him on that day, and on that day deeds are presented to Allah. So he (blessings and peace of Allah be upon him) liked his deeds to be taken up when he was fasting. The fact that he was born on that day was one of several reasons for fasting that day. 

If a person fasts on Mondays, as the Prophet (blessings and peace of Allah be upon him) did, hoping for forgiveness on that day, out of gratitude for the blessings that Allah bestowed upon His slaves on the day – the greatest of which blessings is the birth of His Prophet and His sending him – and hoping to be among the people who are forgiven on that day, then this is something good and is in accordance with what is proven of the Sunnah of the Prophet (blessings and peace of Allah be upon him). But he should not single out one week for that to the exclusion of others, or one month to the exclusion of others. Rather he should do that as much as he can on a permanent basis. 

With regard to singling out one day of the year for fasting it, in celebration of the birth of the Prophet (blessings and peace of Allah be upon him), this is an innovation (bid‘ah) that is contrary to the Sunnah of the Prophet (blessings and peace of Allah be upon him). The Prophet (blessings and peace of Allah be upon him) fasted on Mondays, and no other day, but from year to the next, his birthday may come on that day or any other day of the week.  

For more information on the ruling on celebrating the Prophet’s birthday (Mawlid), please see the answer to questions no. 13810 and 70317. 

Secondly: 

What is widespread among people nowadays of celebrating birthdays is an innovation that is not prescribed in Islam. The Muslims do not have any special days that they celebrate apart from the two Eids, Eid al-Fitr and Eid al-Adha. 

This has been discussed before in several answers; please see no. 26804 and 9485. 

Moreover, what comparison  can there be between the birth of the Prophet (blessings and peace of Allah be upon him), which is indeed a blessing and a mercy for all of mankind as Allah says (interpretation of the meaning): “And We have sent you (O Muhammad صلى الله عليه وسلم) not but as a mercy for the ‘Aalameen (mankind, jinn and all that exists)” [al-Anbiya’ 21:107] and a harbinger of good for all mankind, and the birth or death of any other human being? Moreover, what would that attitude of his Companions, and the righteous who came after them, be towards such an action? 

It is not known that any of the salaf or early scholars said that it is prescribed to fast on any particular day of the week or the month or the year, and regard that day as a “festival” because the Prophet (blessings and peace of Allah be upon him) used to fast on the day of his birth every week, which was a Monday. If that was prescribed, the people of knowledge and virtue of the early generations who hastened to do all that is good would have hastened to do it before us. As they did not do that, it is known that this is something that is innovated and it is not permissible to do it.