Speaking to BBC News NI, Mr McGinn said “everyone who values equality, love and respect can celebrate today”.
“It’s a good day for Northern Ireland, an important day for citizens’ rights across these islands and an exciting day for same-sex couples who can now register to marry,” he said.
Patrick Corrigan from Amnesty International said it was a “historic day for equality and human rights in Northern Ireland”.
“For too long, LGBT+ people in Northern Ireland have been treated as second-class citizens. So, today is an incredible moment for same-sex couples who can finally marry and have their relationships recognised as equal,” he said.
Same-sex marriages have been allowed in England, Scotland and Wales since 2014, but Stormont did not legalise them.
In November 2015, a vote on the issue in the devolved assembly resulted in a numerical majority in favour of same-sex marriage for the first time.