There is nothing unconstitutional about a valid executive agreement.
Unconstitutional, no. Alas, the Constitution doesn't give a great deal of guidance about what does and doesn't constitute a treaty. I don't think statute does either.
But the State Department does have established policy (Circular 175) clarifying what distinguishes an executive agreement from a treaty. They use 8 factors:
(1) The extent to which the agreement involves commitments or risks affecting the nation as a whole;
(2) Whether the agreement is intended to affect state laws;
(3) Whether the agreement can be given effect without the enactment of subsequent legislation by the Congress;
(4) Past U.S. practice as to similar agreements;
(5) The preference of the Congress as to a particular type of agreement;
(6) The degree of formality desired for an agreement;
(7) The proposed duration of the agreement, the need for prompt conclusion of an agreement, and the desirability of concluding a routine or short-term agreement;
(8) The general international practice as to similar agreements.
Furthermore, when the Senate was deliberating ratification of the UNFCCC Treaty in 1992, the Senate asked the Bush (41) Administration if specific emissions commitments made pursuant to the UNFCCC would be put before the Senate for Advice/Consent. And they said that they would.
So, I don't think there's any Constitutional crisis involved with the Obama Administration forgoing Senate ratification of Paris. But I do think it was appropriate for them to have sought it. The closest thing we have to rules delineating Treaties from executive agreements would say so -- but, perhaps more importantly, the governing treaty under which they're operating was ratified with this assurance.