Is a language with only a stack of fixed-size integers Turing-complete?
If your language supports an arbitrarily large array of integers (even of numbers in $\{0,1\}$), then you can simulate the tape of a TM. Then, clearly your language is Turing complete.

EDIT: based on your comment, you don't have an arbitrary array.

In this case, you can do the following:
If you can keep two counters (integers), and increase or decrease them, and check whether they are 0, then you can simulate a two counter machine (Minsky Machine), which is Turing complete (with a caveat).

If you also bound the maximal value that an integer can take, then I believe that your machine is no more powerful than a PDA.