If I were you I would go on the date.
It's hard to act "rude" to a girl you like or want to ask on a date, and your friend was indeed rude to the point where she couldn't handle it and fell out with him. That shows 1) that girl has good standards, and so chances are she is a good person 2) your friend is a p***sy who can't even treat a person right for like what a few days or weeks.
Also, your friend is the one that introduced you to that girl, so in normal circumstances, he should be fine with you two going out. Swearing at you and making you choose between him and that girl is not only childish, but also in a sense "rude". I don't see a point in keeping him as a friend, unless he apologizes for his chidish actions.
Hope this helps in your decision.