I need to ask my partner to watch his language
What do you mean by "watching his language"?
He uses foul language around my family, and it makes them uncomfortable.
I see.  Cursing around others who aren't comfortable with it is rude. You absolutely should speak up to him about things that make you or your family uncomfortable. That way, hopefully, he will understand why he should stop.
But he could claim that it's his free speech, can't he?
It's true that your partner generally does have a right to say what he wants, within certain limits, and foul language is certainly allowed under the law.  However, you should stress that you won't want him around your family if he continues to use that language, because it makes them uncomfortable.  He will surely understand, and try to improve.