It is possible for a deterministic Universe to exist (since a Universe exists and might be determined). It is reasonable to consider the issue a little further perhaps.
One would tend to make an evaluation about the state of pre-destination or determinism in the Universe on the basis of extrapolating from observation and inferences or interpretation of revelation I would think. The Second Law of thermodynamics and the arrow of time seem to support the idea that everything is determined within certain limits. Quantum mechanics and the standard model indicate that all interactions of quanta are pre-determined to occur with certain values and rules of order. Degrees of freedom of motion and of association of quanta have a determined element with an element of uncertainty simultaneously in order to allow structure to exist and change to occur. Being and nothingness are common elements of existence.
“Testimonium divinitatis est veritas divinationis. Tertullian.To foretell things contingent, which depend upon no natural causes, is peculiar to Deity.”
There is another point somewhat interesting; that of different categories of infinites. Cantor and Godel developed a logic of trans-fine sets and of a set of all sets conundrum respectively, yet there may be another transformative sort of infinity, or a different kind of infinity that differs in destination rather than just cardinal or ordinal character. An infinite series of numbers may morph at some point into something else, such as a line of natural numbers morphing into a cloud of numbers, and that may exemplify the space-time or phenomenal importance of location for infinite series. Alternatively an infinite series could just continue, yet as a singularity at T=0 extrapolated from Einstein’s field equations might lead one to contemplate is thought of as reducible infinitely with recursively concentrating gravity, the abstraction of an infinite series might be changed by the physics of quanta and permissible forms of being. A singularity could morph into a field or cloud I guess.