Only rarely will someone - Catholic, Protestant, or otherwise - honestly answer with a, "No, I haven't." More often that not the question is simply ignored and the comments continue.
In one of his parables, he certainly indicates that he will turn people away, those who claimed to have known him but in reality did not (cf. Luke 13:22-30).
While it might be true that Jesus did not explicitly turn anyone way, he nearly did so with the Canaanite woman whose daughter was tormented by a demon (cf. Matthew 15:32-28). It was because of her humble persistence that he answered her request.
He let the rich man go away sad because he wouldn't sell all he owned (cf. Mark 10:17-22). He didn't run after him to comfort him, nor did he change his demands.
He even let the crowds abandon him when he spoke of the necessity of eating his flesh and drinking of blood (cf. John 6:60-71). Once again, he didn't run after them, he didn't change his words to appease them, and he certainly didn't say, "I was only speaking symbolically!" Rather, he turned to the Twelve and asked, "Do you also want to leave?"
Saint Jerome was quite right when he said that "Ignorance of the Scriptures is ignorance of Christ." We cannot claim to know Jesus Christ if we have not read one of the Gospels (they aren't very long, and there are only four of them).
The next time someone claims to know what Jesus would or would not do, or what he would or would not say, without having read just one of the Gospels, I think I'm going to show this image and what for an answer:
Of course, I could end up waiting for a while.