Needless to say, there have been more than a few things said about the current discussion among bishops over 'eucharistic coherence', and whether highly public supporters of abortion access who are Catholic, like President Biden, should be refused communion. There are a few principles involved:
(1) To partake directly of the Eucharist requires that one not merely be Catholic but also have properly confessed your sins recently. Repentance of grave sin is a precondition for communion. The Church has been clear for years that public support of access to abortion, itself a grave sin and a violation of human rights in Church teaching, is a sin that requires such repentance and confession. Strictly speaking, this shouldn't be an issue; politicians who continue publicly to support abortion access should not be going up for communion to begin with. What is more, one of the things you are supposed to do is avoid taking communion if it could cause scandal, skandalon, which in this context means putting a stumblingblock in front of others by misleading them as to what is sinful or not. Yet again, politicians who publicly support abortion access should not be going up for communion at all, because it misleads people as to what Catholic teaching on the subject is. There are plenty of Catholics who, though entirely innocent in themselves, don't go up for communion because they are in some kind of irregular public situation that could cause confusion to others; a relatively common case is of people who are in unusual marriage situations. Politicians don't get special privileged access to the sacraments; they get them only according to the same rules as everyone else. And what is more, it is the Catholic position that partaking of sacraments in ways inappropriate to them is always dangerous for one's own soul.
(2) In unusually public cases that, if ignored, could mislead people as to Catholic teaching, priests and bishops have full authority to prevent people from partaking sacraments. That's their primary function: to uphold Catholic doctrine and to maintain and protect the sacraments. They are not there to be vending machines. Everyone has the right to have their own judgment about whether their decisions are reasonable and prudent; all Catholics have the right to write a letter to their own bishop and give their reasons for why they think it a bad idea, if they do. But there is no question whatsoever that they have the right and authority to make such a decision.
(3) The Eucharist is a public sacrament; that is, one can participate indirectly in it simply by being prayerfully present at Mass. Denying someone communion, which implies affirmation of the faith and morals of the Catholic Church, is not (as I saw someone claim) denying them the real presence of Christ. Precisely because it is the real presence, anyone has access to it just by going to Mass or Adoration, without receiving.
A great deal of what one finds on the subject is much like what we get with this Jeffrey Salkin article. Salkin (who unlike a number of people I have seen is at least upfront about the fact that he is not Catholic, and avoids most of the more stupid arguments) basically gives three arguments for why he thinks denying Biden communion is a bad idea.
(1) "[I]t seems foolish to deny access to the body of Christ to the most visible Catholic political leader in the world." This is the reverse of the actual situation; the only reason that the question even arises is because he is so visible and people like Salkin keep trying to hold him up specifically as a Catholic. In other situations, it wouldn't even arise (the individual would just have to answer to God, or, if it were a definitely public situation, it would be between the individual and their priest or bishop). It's mostly an honor system. The only situation in which this would even be on the table is in a very public and visible case, in which scandal is a serious danger. As the most visible Catholic political leader in the world, President Biden has an even greater responsibility not to act in a way that would mislead people about Catholic faith and morals.
(2) "[T]his cannot be good for the church — not at a time when it has experienced profound losses." I confess myself wholly baffled by this kind of argument. Why would anyone care about something like this? The sacraments are part of the integral structure of the Church; decisions about them should be based on considerations about whether they are in danger of being abused or in danger of being received in a manner inconsistent with Catholic faith and morals. Nothing else matters on a point like this, and it is absurd to expect anything else to matter on something like this.
(3) "[T]he denial of Communion to President Biden smacks of inconsistency at best and hypocrisy at worst." This could very well be, although Salkin's argument is not very good. (Sexual abusers are not supposed to receive communion, either, until they repent of their sins, and this is already known.) What raises the question here is a prominent Catholic engaging in very public and continuing support of what the Church regards as a sinful practice on a matter on which the Church has already been quite clear. This is not, contrary to Salkin's implication, a common situation. It is, in any case, irrelevant; while obviously consistency would be better, the exact lines are a matter of judgment call, and it is the judgment call of the bishops as to where they will be laid. You can think them wrong; it's certainly true that these are days in which we are repeatedly faced with bishops who are neither very saintly nor very intelligent. But that they aren't following some strict rule in handling these matters is irrelevant because there is no such rule, and however one may impugn their motives, it is irrelevant because there is no doubt at all that it is their call, and theirs alone to make. And, frankly, I find you can always tell when someone has no serious argument on a religious topic when they start accusing people of hypocrisy, an accusation so over-used that it has by now almost reached the point of self-parody.
My suspicion is that this has mostly come up just because the bishops are tired of Biden's Catholicism being used as political cover by the media. Of course, here as elsewhere, it will be the individual bishops who will decide how the rule should be enforced; people have a bad habit of thinking that rules magically go into effect on their own, but in reality, they have to be operationalized into real situations. There are bishops who would likely still allow him (and other Catholic politicians in similar situations) to receive, despite the majority, and how publicly will likely depend on what they think Rome would let them get away with. We will see if anything really comes of it.