I’m a Muslim girl from Romania. How can I get a visa to visit the Prophet’s Mosque in Medina, Saudi Arabia?
Sorry, you can't. Women under 45 are not granted pilgrimage visas unless they travel with a male guardian, and even if you're over 45, you need to travel with a group. Royal Embassy of Saudi Arabia to the US:
4) All women are required to travel for Hajj with a Mahram [male guardian, see Wikipedia]. Proof of kinship must be submitted with the application form. Women over the age of forty-five (45) may travel without a Mahram with an organized group, They must, however submits a no objection letter from her husband, son or brother authorizing her to travel for Hajj with the named group. This letter should be notarized.
See also: How to independently visit Saudi Arabia?