Larousse and Collins both refer to gémir for groaning (of pain). Saying that gémir implies weak/high-pitch sounds is not entirely accurate:
Pousser un, des cri(s) étouffé(s) et plaintif(s) exprimant une douleur ou un malaise physique.
A gémissement could be faible (weak), long, aigu (high pitched), étouffé (muffled), profond (deep), sourd (dull), rauque (hoarse), douloureux (painful), or lamentable (plaintive) (TLFi). The grognement (grogner) is associated with the sound the pig, the wild boar, the bear, and by extension the dog, or even the hedgehog, make (Larousse, TLFi) i.e. growling. It can be used by analogy for a human being making dull, muffled, thudding, or generally inarticulate sounds : « Son mari grognait de douleur dans son sommeil. » (Zola). With gémir that reference to pain is also common whether that's redundant or not. See Books (ngram) for further contextual cues (including cri de mort, which I know from I know not where).
As another answer points out, groan doesn't originate from the belly, if but metaphorically speaking, but rather from Proto-Germanic *grain-, by imitation or related to grin, and with the Old Norse cognate grenja "to howl" (Etymonline). To groan is neither to growl nor to howl, but hurler in French most definitely has the high-pitch association (dog, wolf) and meaning (Pousser des cris aigus et prolongés, TLFi) compared to gémir. But because of polysemy and varied usage scenarios, personally I wouldn't restrict my choice of verb/noun (crier/gémir/hurler/grogner/cri/gémissement/hurlement/grognement) based on pitch perception but rather usage and context, such as manner, intensity and cause etc. Cri, plainte, pleur, râle, sanglot, soupir are mentioned in the TLFi entry for gémissement as being in the same sort of semantic field and can prove useful depending on context. When one wants to avoid using adverbs which may feel less casual in a verb construction, you can resort to using another verb to introduce the noun with the adjective instead, in particular pousser as in pousser un cri/un gémissement plaintif for instance, instead of gémir plaintivement. Finally, when one discusses anger rather than pain, there is tonner which is really used for some sort of angry rant yet carries that thunderous (i.e. tonnerre) sort of idea you refer to with gronder which cannot be used in this fashion with a person nowadays (could still be said of an animal though).
The Q&A shows a myriad of verbs and adjectives with seemingly endless possible combinations where one can fine tune the property of the sound such as its (low) pitch etc., which is really the opposite of what the question title implies. The gémissement is not restricted to a high-pitched sound, unlike with the base meaning for hurlement, or weak per se; it is not a mélopée. One can have a construction with [pousser + un cri/gémissement + adjective ex. sourd] instead of [verb + adverb] if one so requires. Gémir de douleur doesn't appear that far off the mark, bar a very specific context (which was not provided). Satisfaction is a matter of opinion but I would say I'm quite satisfied with the rich semantics and lexicon, and then some...
Il faut se battre! gargouilla la voix rauque d'un corps qui, depuis
notre réveil, se prétrifiait dans la boue dévoratrice. Il le faut ! —
Et le corps se retourna pesamment. [...]
[ H. Barbusse, Feu, in the TLFi at gargouiller, inspired from a prior edit to another answer ]