Is there a problem solution which, if implementable in a given language, implies that that language is Turing complete? I have been researching a bit lately and found myself thinking about the title question for some time now, but have found nothing conclusive.

For example, some problems require loops practically - making 10^100^1000^10000 repeated calculations (even no-ops) is not possible practically as the space necessary to store the code would likely be greater than the number of atoms in the universe, but would be possible theoretically given infinite memory (similar to how current implementations of programming languages only approximate Turing-completeness).

Alternatively: is there a problem which requires Turing-completeness to solve theoretically?
The answer is no (although from a rather syntactic reason, which might not be what you were looking for).

Let $f : \Sigma^* \rightarrow \Sigma^*$ be some arbitrary function, then it is implementable in the following "stupid" language which is not Turing complete:

Any string $s\in \Sigma^*$ is a valid program, and all strings represent the program which, given $x$ as input, outputs $f(x)$. Note that I don't really care how it computes $f$ (as I'm defining a new model of computation). If this bothers you, imagine any string is translated to the same fixed Java program which computes $f$ (assuming $f$ is computable).