No, you cannot cast a (ritual) spell during a short rest.
At least, that's the implication this GM gleans from the rules as written. Vexingly, the key the Long Rest rules have some of the relevant text, though even there it's awkward. Specifically, the Long Rest rule states:
If the rest is interrupted by a period of strenuous activity - at least 1 hour of walking, fighting, casting spells, or similar adventuring activity - the characters must begin the rest again to gain any benefit from it.
That is: the Long Rest rules give us a minimum set of activities which count as "strenuous". Unfortunately, it's unclear - grammatically - whether "1 hour" applies just to "walking" or also to "fighting, casting spells, or similar adventuring activity".
This GM is of the opinion that "1 hour" applies just to "walking". I believe that is in agreement with the other rules text about rests and the fact that rests are meant to be, well, restful - you're not supposed to be doing anything in particular (with a carve-out for a 2-hour watch for long rests). Casting a spell - whether with spell slots, using an innate ability, or as a ritual - is just as active (ie., "strenuous") as swinging a sword at an enemy.
From a theoretical perspective, this GM sees the option of either "take a rest and gain whatever benefits that offers" or "cast some ritual spells and gain whatever benefits those spells grant"; being able to cast (ritual) spells during a rest results in double-dipping.