Can I get my marriage to a non-Christian blessed by the Catholic Church?

If there are no other impediments to the marriage, you can—and indeed are obliged to—have your marriage convalidated by the Church. As a Catholic, you are under Church law and your marriage must be recognized by the Church in order to be valid. Once your marriage is convalidated, you will have a valid union (though not a sacramental one, since those who are not baptized cannot participate in the other sacraments). See your local priest about having your marriage convalidated. It should be a fairly simple thing to do.