No.
No, no, no, no, no.
At least not in normal D&D.
For that matter, it'd be a hard sell to try and make any claims that a woman couldn't be a fighting class no matter what the campaign setting. Even if there's a strong patriarchal mindset to the game setting, saying a woman couldn't be a fighter is still impossible - there's always going to be a few exceptions here and there, though people who meet them might show a great deal of bias if they show off their skills or possibly even try to kill them. But that's something that should be dealt with during the game.
And women not being able to be paladins?
My definition of Good doesn't include sexism in it.
So yes, women can be paladins. (Or men, too. I'm looking at you, Scarred Lands...) It's a warped point of view that a god embodying Good would allow sexism.
There might be more to this, but from the sound of it, your DM is just being a prig.