Of course not!
And yet some people -- who apparently have either not read Aquinas or not read him carefully -- persist in thinking that he holds that God (along with creatures) is a participant in being. You can find this, for example, in some Barthians (and perhaps even in Barth himself).
Consider this passage from Angela Dienhart Hancock's recent book on Barth:
The analogia entis, a scholastic term famously formulated by Thomas Aquinas, means that God and human beings (and all creation) are similar in that they participate in something called “being.” Hence we can figure out what God is like by looking at the created order, because everything is connected to everything else in the great chain of being, which stretches all the way up to being itself (God) (Karl Barth’s Emergency Homiletic, 1932-1933: A Summons to Prophetic Witness at the Dawn of the Third Reich, p. 186, n. 142).
Three things. First, I have never found the term analogia entis in Aquinas's writings. I do not deny that the idea is there formally (although not in the way suggested by Prof. Dienhart Hancock). I only say that I have not been able to find the term itself. Second, if God is being, as is suggested at the end of the above passage, how can he participate in being? Third, the first sentence, as a statement about Aquinas's doctrine is simply false.
There are many places where Aquinas makes his doctrine clear. One such place is this passage from the commentary on the Sentences:
[C]reator et creatura reducuntur in unum, non communitate univocationis sed analogiae. Talis autem communitas potest esse dupliciter. Aut ex eo quod aliqua participant aliquid unum secundum prius et posterius, sicut potentia et actus rationem entis, et similiter substantia et accidens; aut ex eo quod unum esse et rationem ab altero recipit, et talis est analogia creaturae ad creatorem: creatura enim non habet esse nisi secundum quod a primo ente descendit: unde nec nominatur ens nisi inquantum ens primum imitatur; et similiter est de sapientia et de omnibus aliis quae de creatura dicuntur (Prol. q. 1, a. 2, ad 2). (The Creator and the creature are reduced to unity not by a univocal commonality but by analogy. Now, such a commonality can be twofold. It can either be because certain things participate in something according to an order of priority and posteriority, as potency and act participate in the concept of being as also substance and accident do; or this commonality can be because a thing receives its being and its concept from another, and such is the analogy between creature and Creator. In fact, the creature only has being insofar as it descends from the first being [i.e., God]. Thus, it is only called being because it imitates the first being. And the same must be said of wisdom and everything else that is said of creatures.)
What more needs to be said?