No, the Buddha didn't say women can't become fully enlightened. There were many fully enlightened Bhikkhunis in the Buddha's day. Look up Patacara for example, she was amazing.

There's just one sutta in the Majjhima Nikaya which says a woman can't become a Buddha. Some research suggests it may be a later addition. But if it isn't, it's probably because people wouldn't listen to a woman. Because human societies are almost always patriarchal. That would also explain why the same sutta says a woman can't be Mara.

Being a Buddha and being fully enlightened aren't the same thing. All Buddhas are fully enlightened. But not all fully enlightened beings (arahants) are Buddha's.

There is no living tradition of Buddhism that holds the belief that women can't become enlightened.