Sara, you really need to talk to her friend's parents! What if your daughter was lactose intolerant or had a milk allergy and he was still trying to convince her to drink milk? I think you need to talk to the parents about your dietary choices and what your daughter's diet is like before you let her go over to play again. You don't have to be mean or threatening about it, but just explain why you live the way you do and tell them that the grain and nut milks she drinks at home have calcium, vitamin D, vitamin A, just like milk does (as well as any other nutrient issues). They might just be concerned about her welfare, but if you explain to them nicely, they should be fine with your decision and will stop offering her animal foods and drinks. I mean, what if he was trying to offer her chicken nuggets or something?
Basically, if they can't respect your wishes, simply don't let your daughter go over to their house anymore. You shouldn't worry about starting trouble with them, because it sounds like they started the trouble first by trying to get her to drink milk when your daughter probably told them she shouldn't/couldn't drink it.
You might also find that keeping your daughter at home during meal times and snack times will make it so she doesn't want to eat/drink what they have. If she's full and not thirsty all she has to say is that she's eaten already or not hungry/thirsty.