No, there is not. See https://en.wikipedia.org/wiki/Complete_theory . For example, Presburger arithmetic (with only addition and not multiplication) is not incomplete, and Gödel's incompleteness theorem does not apply to it.
Gödel's incompleteness theorem arises because arithmetic is capable of modeling itself - within itself - which allows the creation of a Gödel sentence. Presburger arithmetic isn't powerful enough to model itself. Instead of saying Gödel's incompleteness theorem applies to "any formal system powerful enough to model arithmetic," we could say it applies to "any formal system powerful enough to model itself." If a formal system can model arithmetic it can model itself, because arithmetic is powerful enough to model any formal system. (We might very informally say that this is because arithmetic is "Turing-complete"; it is strong enough to work like a computer program, and for any formal system, we can write a computer program to check proofs in that system.)