I would say yes.
Here are some supporting points.
1. Did Almighty Allah honour anyone's birthday?
وَسَلاَمٌ عَلَيْهِ يَوْمَ وُلِدَ وَيَوْمَ يَمُوتُ وَيَوْمَ يُبْعَثُ حَياً
"And peace be on him the day he was born, the day he will die and the day he will be raised" (Qur'an 19:15)
So Allah acknowledges the birth of Isa (19:33) and Yahya (as) saying clearly peace on 'the day he was born'.
Allah acknowledges the blessed day of birth. When we celebrate someone's birth, we call it a 'birthday'.
2. Did the companions hold a gathering in honour of our Prophet (ﷺ)?
It was narrated that Abu Sa'eed Al-Khudri said:
"Mu'awiyah (ra) said: 'The Messenger of Allah (saw) went out to a circle of his Companions and said: 'What are you doing?' They replied, 'We have come together to pray to Allah and praise Him for guiding us to His religion, and blessing us with you.' He said, 'I ask you, by Allah, is that the only reason?' They said, 'By Allah, we have not come together for any other reason.' He said, 'I am not asking you to swear to an oath because of any suspicion; rather Jibreel came to me and told me that Allah is boasting of you to the angels.'"
أَخْبَرَنَا سَوَّارُ بْنُ عَبْدِ اللَّهِ، قَالَ حَدَّثَنَا مَرْحُومُ بْنُ عَبْدِ الْعَزِيزِ، عَنْ أَبِي نَعَامَةَ، عَنْ أَبِي عُثْمَانَ النَّهْدِيِّ، عَنْ أَبِي سَعِيدٍ الْخُدْرِيِّ، قَالَ قَالَ مُعَاوِيَةُ رَضِيَ اللَّهُ عَنْهُ إِنَّ رَسُولَ اللَّهِ صلى الله عليه وسلم خَرَجَ عَلَى حَلْقَةٍ - يَعْنِي مِنْ أَصْحَابِهِ - فَقَالَ " مَا أَجْلَسَكُمْ " . قَالُوا جَلَسْنَا نَدْعُو اللَّهَ وَنَحْمَدُهُ عَلَى مَا هَدَانَا لِدِينِهِ وَمَنَّ عَلَيْنَا بِكَ . قَالَ " آللَّهِ مَا أَجْلَسَكُمْ إِلاَّ ذَلِكَ " . قَالُوا آللَّهِ مَا أَجْلَسَنَا إِلاَّ ذَلِكَ . قَالَ " أَمَا إِنِّي لَمْ أَسْتَحْلِفْكُمْ تُهَمَةً لَكُمْ وَإِنَّمَا أَتَانِي جِبْرِيلُ عَلَيْهِ السَّلاَمُ فَأَخْبَرَنِي أَنَّ اللَّهَ عَزَّ وَجَلَّ يُبَاهِي بِكُمُ الْمَلاَئِكَةَ
The companions gathered in honour of the Prophet (saw) thanking Allah for blessing us with you. This was witnessed by him and acknowledged by Allah and appreciated by the Angels. That is what a meelad gathering is - to thank Allah for sending the Prophet and to honour and praise him.
3. Did the companions ever celebrate anyone's birthday?
Mu'awiya ibn Qurra said, "When Iyas was born to me, I invited a group of the Companions (ra) of the Prophet(saw), and I fed them and they made du'a. I said, 'You have made du'a, so may Allah bless you for your du'a. If I make du'a, then please say, "Ameen".' He went on, 'I made a lot of du'a for him for his deen, his intellect and things like that.' He added, 'I still recognise the du'a of that day in him." (Al-Adab al-Mufrad, Imam Bukhari, Saheeh)
سَمِعْتُ مُعَاوِيَةَ بْنَ قُرَّةَ يَقُولُ: لَمَّا وُلِدَ لِي إِيَاسٌ دَعَوْتُ نَفَرًا مِنْ أَصْحَابِ النَّبِيِّ صلى الله عليه وسلم فَأَطْعَمْتُهُمْ، فَدَعَوْا، فَقُلْتُ: إِنَّكُمْ قَدْ دَعَوْتُمْ فَبَارَكَ اللَّهُ لَكُمْ فِيمَا دَعَوْتُمْ، وَإِنِّي إِنْ أَدْعُو بِدُعَاءٍ فَأَمِّنُوا، قَالَ: فَدَعَوْتُ لَهُ بِدُعَاءٍ كَثِيرٍ فِي دِينِهِ وَعَقْلِهِ وَكَذَا، قَالَ: فَإِنِّي لَأَتَعَرَّفُ فِيهِ دُعَاءَ يَوْمِئِذ
It is clear that birthdays were celebrated with food and dua's by the companions (ra). These dua's were made in congregation. No companion refused to attend, refused to eat or refused to make du'a. So the companions believed in birthday celebrations as does Imam Bukhari who records this.
4. Did the Prophet (saw) commemorate his own birthday?
Abu Qatada Ansari reported that Allah's Messenger (saw) was asked about fasting on Monday, whereupon he said: It is (the day) when I was born and revelation was sent down to me. (Muslim)
The Prophet (saw) fasted on his weekly birthday and he remembered the day of his birth regularly via a religious act. So, we should also remember it regularly with a religious act as it's his sunnah.
This appears in Muslim no less than 4 times, emphasising that honouring his birthday on its weekly anniversary is sunnah. Thus it is sunnah to regularly honour his day of birth as the Prophet (saw) made no restriction by limiting it to weekly only, so we have no right to restrict. Weekly, monthly, annually are all permitted. Nowhere is such a gathering forbidden or restricted.
5. Did the Prophet (saw) ever attend his own gathering in honour of his birth?
The Prophet (saw) went with Abu Darda to the house of Amir Ansari and found him to be talking about the events of the birth of Prophet (saw) to a gathering of his friends and children and was repeating 'this was the day' (i.e. 12th Rabi-ul awwal). The Prophet (saw) stated, "O Amir Allah has opened the doors of his mercy for you and the Angels are praying for your forgiveness. Whoever does this act of yours, he would also get the salvation like yours". (Imam Suyuti)
6. Are we required as Muslims to celebrate our children's birthdays:
Aishah (r) narrated that the Allah's Messenger (saw) commanded us to sacrifice two sheep for a boy's Aqeeqah and one sheep for a girl." (Ibn Majah, Hasan)
عَنْ عَائِشَةَ، قَالَتْ أَمَرَنَا رَسُولُ اللَّهِ ـ صلى الله عليه وسلم ـ أَنْ نَعُقَّ عَنِ الْغُلاَمِ شَاتَيْنِ وَعَنِ الْجَارِيَةِ شَاةً
We are required to sacrifice and distribute food to celebrate the birth of our children. This is not un-Islamic - it is Islamic. Even those who reject birthdays do this celebration of birth.
7. Are Muslims required to honour other birthdays of other Prophets?
One of the reasons Friday is blessed is because it represents the creation of Adam (as). He was not born but created and created on a Friday and Allah has blessed that day as a result.
There are many more evidences, but these should be enough for anyone who is sincere.
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