As Geoff points out, the answer is likely to be no, though it takes some work to apply the known finite group theory here. It should be emphasized in the question that the group considered is *finite* as indicated by the tag 'finite-groups', so I edited the question and added another tag. (The question would also make sense in the more open-ended setting of infinite groups, where there are lots of Coxeter groups whose products of involutive generators have finite order---but no clearcut classification.)

It's also essential here to realize that the property of being a "Coxeter group" depends on specifying particular generators: an abstract group may be a Coxeter group in more than one way. Also, it's important to distinguish between "irreducible" Coxeter groups and direct products of smaller ones. Finite irreducible Coxeter groups have a nice classification: they are either crystallographic (Weyl groups of simple Lie algebras) or among the remaining dihedral groups along with two extra reflection groups called $H_3$ and $H_4$.
In all these cases one has good information about the internal structure (Sylow theory, etc.) and the character tables.

To exhibit a very small example of a group $G$ satisfying your kind of conditions but not a Coxeter group, probably the best you could do is to start with the Weyl group $W$ of type $D_4$ and factor out its center (of order 2) to get $G$. Here $G$ has order 96 and like $W$ has subgroups isomorphic to $S_4$ of order 24. So by the classification, $G$ couldn't be one of the *irreducible* Coxeter groups. Since $G$ is solvable (though not nilpotent), it has plenty of normal subgroups. To rule out a decomposition of $G$ as direct product of two proper normal subgroups, you'd have to do more work using the known characters of $W$ or perhaps the Sylow structure of $W$. Though I haven't done all the work at this point, I agree with Geoff that there should be counterexamples.

Finally, I'd suggest that the only reasonable necessary-and-sufficient condition for a finite group generated by involutions to be a Coxeter group is that the other relations involving orders of products of generators be the sole relations one has to add to define the group. Short of this, I'm not sure whether there are any useful necessary-or-sufficient conditions.