"Give a man a fish, and you feed him for a day. Teach a man to fish, and you feed him for a lifetime" is a proverb. And a proverb is a kind of idiom. We usually don't change words in idioms because those are set expressions. In this particular proverb it may also be necessary to keep "fish" in the second part in order to contrast the first fish and the second - a fish vs. to fish. So, I think it's important to keep it as it is for a better effect, for emphasis, if you will.
In addition, we usually (but not always) say "to teach someone (how) to do something" (Macmillan dictionary):
My uncle is going to teach me to drive this summer.
(but "His mother had taught him some words in Spanish." (note that "words" is a noun, while "drive" is a verb in the sentence above)
All in all, I strongly recommend not to use "fishing" instead of "fish" in the proverb.