Those White House website petitions are the equivalent of liking something on Facebook.
They're probably even less effective. I've yet to hear of any sort of legislation come about as a result of one of these. I guarantee that if people were actually required to make so much effort as to write something on a piece of paper, put it in an envelope, and stick it in the mailbox, this petition (and all the others on that site) would have a scant fraction of the signatures it does now.
And what good is it, anyway? We know his stance. He can't force one through without the House. It's just a self-aggrandizing episode of preaching to the choir.