Abrogation of a treaty is not listed in the Constitution, but understood[0] to be the President's right, just as it is the Prime Minister's right (see "Crown prerogative") in the UK. As a matter of fact, presidents have done this[1], and by and large it has gone unchallenged, and subsequent Presidents and Congresses have not restored abrogated treaties either.

Arguably the ability to abrogate a treaty is very important, as it balances the ability of a treaty to impose considerably on the laws of the U.S., as the Constitution and the treaties are the supreme law of the land, but treaties do not require anywhere near as much review and approval as Amendments! The President's ability to abrogate a treaty is a safety device that allows quick remediation of unjust/tyrannical/ultimately-undesirable treaties. (Apparently, Congress too can abrogate treaties. See below.)

I would be shocked and surprised if the SCOTUS were to rule that the President cannot alone abrogate a treaty. The Constitution simply says nothing about the topic, and thus it would seem like a political question (that's a term of art meaning: the courts do not meddle) to be decided by Congress and the President, and/or any Constitutional Conventions the States might call. The SCOTUS would have to invent an abrogation method out of thin air in order to rule that POTUS can't alone abrogate a treaty. EDIT: More than that, the SCOTUS would have to then explain whether and why previously abrogated treaties are to be considered restored or not. The SCOTUS doesn't like to be in such sticky situations, even when the majority of Justices might greatly prefer the alternative outcome on the specific policy issue before them.

Assuming the presidential power to abrogate, it doesn't matter what the UNFCC says, if POTUS wants to abrogate it, that's that, and he can either abide or not by the withdrawal process -- once a treaty is officially abrogated, its withdrawal clauses no longer have force on the country in question. The same applies to the Paris accord. Trump could leave effective immediately rather than with the accorded three-year period, and what could the other nations do? In theory they might apply sanctions (EDIT: even go to war!), but in practice it's not currently politically feasible for any significant powers to apply any meaningful, and maybe even merely symbolic, sanctions.

In [1] there is a list of treaties abrogated by the U.S. -- it's a bit out of date now, but not by much. Skimming it on my phone I counted some seven instances (careful with the asterisk'ed one) when the President abrogated a treaty without Congressional approval, once at least in spite of Congress refusing to repudiate the treaty. Another time Congress approved after the fact; I did not count that one. And we have the ABM treaty abrogation to add to that list. And who knows, perhaps soon the UNFCC as well. A number of times Congress has repudiated treaties by resolution, others by law. One treaty was "suspended", but presumably never restored, thus effectively abrogated. None of that even counts treaties that we do not comply with without saying that we won't comply with them (I'm sure there are some, perhaps many, such treaties, mostly out of obsolescence or carelessness).

Given the make-up of Congress at this time, it seems exceedingly unlikely that it would act to prevent abrogation of any treaty by POTUS. Perhaps in 2018, 2020, or later, but not at this time, and for obvious reasons. Given the attempts by the previous POTUS to bind the hands of the current POTUS, I would not be surprised if the next POTUS (if a Democrat) might not attempt to restore by fiat any treaties Trump might abrogate by fiat, but I doubt it. Recall that Clinton did not bother sending Kyoto to the Senate for ratification, and that 96 Senators had indicated their opposition to it -- assuming the Democrats win big in 2020, it's not safe to assume they would restore the UNFCC were Trump to abrogate it, or the Paris accord, as once they have won their election they won't have incentive to do things that they might actually consider against their interests (and they might this). Still, given the hazy status of abrogation in Constitutional Law, if a future administration restores an abrogated treaty, I expect such a decision to be adhered to and not require re-ratification by the Senate or any Congressional approval of any kind. After all, to abrogate a treaty is merely to stop obeying it, and whether abrogation be intentional or not, surely to abide once more by the abrogated treaty must be a perfectly normal (if unheard of) thing to do as it is merely curing an unofficial state of non-compliance.

[0] http://caselaw.findlaw.com/us-supreme-court/444/996.html
[1] http://www.ea.sinica.edu.tw/QDown.ashx?ID=5563