Does Islam permit same-sex marriage?

No. It's haram i.e. strictly forbidden.

It is further worth noting that homosexual relations are considered a death penalty offense in Islamic Law, so recognition of same-sex marriages in Islam would be seen as giving sanction to illegal conduct.

The nation of Hazrat Lut (AS) was annihilated from the face of the earth for homo-sexuality. It is un-natural, illogical, an act of perversion, and an activity that invites the wrath of Almighty Allah. Islam religion is based on nature. Any action against the laws of nature is prohibited in Islam.