Having skimmed through the earlier posts; I agree that we are not to pass judgment on our fellow men, nor are we to usurp the position of the bishops of the pope in determining what is moral. I also agree that we do not know exactly what Jesus would do because we do not know Him perfectly, and also because we do not know the interior state of the prospective communicants’ hearts.
Having said this, I will now pass judgment.

While I respect the bishops who have allowed publicly dissenting people to receive communion may have the authority to do so (which authority I respect) and may have good reason to do so, I certainly have the right to point out what I think may be an erroneous decision. I obey the law; I can still say I think some laws are applied poorly.
When people ask “What do you think Jesus would do” I get the feeling that what they really mean is “Jesus is a welcoming loving nice guy God who would never turn other people away so basically let people do what they want Jesus will take care of them anyway besides what business is it of yours?”
Of course, the first part is true. Jesus turns no one away.
However, I think the question of what Jesus would do is largely irrelevant. I think the real question to ask is: if Rudy Giuliani met Jesus, what would he do? Because that may be no business of mine, but it is definitely business of his.
The fact that Giuliani (or any other publicly and gravely dissenting Catholic - by the way that is not meant to be a judgment, simply a label for his/their public position) can reject a core Church teaching on abortion on one hand yet fully accept that he is accepting the Real Presence on the other seems to me to be a great contradiction. I would instead suspect that he is unaware of the tremendous gift Christ gives of himself in the Eucharist.
Perhaps to him it is an important tradition or way of life. Perhaps he is unsure of where he stands on the issue of abortion. Perhaps it is a useful political tool to be seen in Church. I hardly know.
But if he really believed that he was meeting Jesus, in person, mano a mano (mano a panno? sorry

) every Sunday, he could not continue to publicly support a grievously sinful and morally (and literally) fatal issue. Refusing communion might be essential for his very salvation, reminding him of his serious obligation to respect the Divine Person.
What would Jesus have done? Well, I seem to remember that he turned the moneylenders out of the Temple, rather violently. I seem to remember that he let the rich man, perfect in observance of the Law, go away because he could not let go of his wealth. Yes Jesus welcomes everyone, but he also confronts us with a choice. “Do you know who I am?”, “Are you willing to give up those things that will ultimately keep you from accepting My love?”
I don’t think that Jesus was angry because the moneylenders were sinful people (He ate with prostitutes, thieves and tax collectors). I think He was angry because they had become oblivious to their presence in the very house of God, and He needed to remind them sharply of that. The scandal was not their rejection of God or the Law. Their scandal was worse, a complacently that would make God and his Law appear irrelevant.
Refusing communion to Giuliani would not necessarily convince him to change his ways, nor would it make the Bishop popular. But it would remind the faithful not only of their obligation to accept central moral Church doctrines, but of the significance of receiving the Eucharist.
Giuliani, like all of us, has to face those tough choices and pick a side of the fence to sit on. Better for him to make a choice now, while he still has time to repent. If his choice would require him to.