Short answer
Yes. Visiting the grave of the Prophet ﷺ is one of the most rewarding acts for anyone who can reach Madinah, and Hanafi scholars call it close to wajib for those able. The four schools agree it is recommended. Scholars differ only on whether to make the grave, or the mosque, the purpose of the journey.
Ziyarah of the Prophet's ﷺ grave is among the best recommended acts, described by Hanafi jurists as close to wajib for one with the means. It is not a part of Hajj. The visitor greets him with respect and a lowered voice, then greets Abu Bakr and Umar.
The four schools agree on this.
Explanation
The Prophet ﷺ said that no one greets him with salam except that Allah returns his soul to him so that he returns the greeting. He also allowed visiting graves in general to remember the hereafter. Muslims through history have greeted him at his grave, and the four schools agree this visit is recommended. See also is visiting Madinah part of Hajj.
The point of difference: the Prophet ﷺ said journeys are made only to three mosques: the Sacred Mosque, his mosque and al-Aqsa. Some scholars, notably Ibn Taymiyya and those who follow him, read this as forbidding a journey made for the sake of a grave, though they agree the visit is sunnah once you are there for the mosque. Hanafi and most other scholars read the hadith as about mosques only, and allow travelling with the grave in mind. In practice everyone intends both the mosque and the greeting, which settles it.
- How: pray two rakats in the mosque first, then go to the grave, stand facing it at a respectful distance and say: "As-salamu alayka ya Rasulallah", then greet Abu Bakr and Umar.
- Voice: low. The Quran warns against raising voices above his.
- Avoid: touching or kissing the grille, bowing, circling it, or praying towards it.
- Dua: ask Allah directly; many scholars say face the qiblah for it.
The reports that "whoever visits my grave will have my intercession" are graded weak by many hadith scholars, so we do not rely on them here. The authentic hadith above is enough. While in Madinah, see Riyad al-Jannah and Masjid Quba.
The evidence
Quran, Surah Al-Hujuraat 49:2
يَٰٓأَيُّهَا ٱلَّذِينَ ءَامَنُوا۟ لَا تَرْفَعُوٓا۟ أَصْوَٰتَكُمْ فَوْقَ صَوْتِ ٱلنَّبِىِّ وَلَا تَجْهَرُوا۟ لَهُۥ بِٱلْقَوْلِ كَجَهْرِ بَعْضِكُمْ لِبَعْضٍ أَن تَحْبَطَ أَعْمَٰلُكُمْ وَأَنتُمْ لَا تَشْعُرُونَ ﴿٢﴾
O you who have believed, do not raise your voices above the voice of the Prophet or be loud to him in speech like the loudness of some of you to others, lest your deeds become worthless while you perceive not.
Respect and a lowered voice in his presence.
Sunan Abi Dawud 2041
Grading to be checked... ما من أحد يسلم علي إلا رد الله علي روحي ...
"No one greets me except that Allah returns my soul to me so that I return his greeting."
The virtue of greeting the Prophet.
Sahih al-Bukhari 1189
Sahih... لا تشد الرحال إلا إلى ثلاثة مساجد ...
"Journeys are not made except to three mosques."
The hadith at the centre of the scholars' difference about travel.
Where the ruling is recorded
- Radd al-Muhtar (Ibn Abidin), Kitab al-Hajj, matlab fi ziyarat qabr al-nabi. The visit is close to wajib for one with the means; its etiquette.
- al-Ikhtiyar, Kitab al-Hajj, fasl fi ziyarat al-nabi. The manner of greeting the Prophet and his two companions.
- Fath al-Qadir, Kitab al-Hajj, at the end. Ibn al-Humam on making the intention of the visit.
Checked by our team. For a more accurate answer to your own situation, consult a scholar. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.