Is simply typed lambda calculus with fixed-point combinator Turing-complete?

The simply-typed $\lambda$-calculus with the fixpoint combinator but without a primitive integer type is not Turing-complete, at least not in the usual sense (Church integers and computation as $\beta$-reduction). This is a consequence of a result of Statman [1], stating that (somewhat surprisingly) termination in that calculus is decidable.

Perhaps what you want is the simply-typed $\lambda$-calculus with a fixpoint combinator and a primitive integer type, together with its constructors (zero and successor) and destructors (ether a single "case" destructor, or if-then-else and predecessor). This is known as PCF and showing that it is Turing-complete is a straightforward programming exercise (I don't think there is a reference for it in the literature).

[1] Richard Statman. On the lambdaY calculus. Ann. Pure Appl. Logic 130(1-3): 325-337 (2004)

Damiano is right, the answer is no: it is not Turing complete.

Supposing to define natural numbers as some type (o->o)->o->o, even with fixpoints you cannot define the predecessor (that would be enough to have equality, and then, by Goedel characterization via mu-recursion, all computable functions).

In fact, by just adding fixpoints you only get partial polynomials instead of the total polynomials of the traditional calculus. This can also be seen by generalizing Schwichtenberg approach (see e.g. Functions Definable in the Simply-Typed Lambda Calculus) Instead of normal forms you must consider bohm-trees, now, that have very limited shapes due to typing constraints. Head variables in terms are the same as in the traditional case, and the proof proceeds in a very similar way.