I was practicing my full golf swings yesterday, and I am now experiencing some mild pain on my left wrist. Should I just leave my wrist alone?

Rest, ice. And an Ibuprofen or two 2-3 times a day. No better 3-5 days please see your doc.
Yes. You probably have a minor sprain. You can ice it, take some antiinflammatories like Ibuprofen or naproxen. But most importantly, don't try golfing again while it's still hurting. You need to let it heal first.