It is a bit unclear what you would mean by this precisely which is a bit of a downside of the question. Let me sketch some very rough sort of answer to highlight the issue: Euler considered the zeta function at integral values (only), then gamma is the factorial (or at least one can write it out in a more explicit way). So in some sense the question disappears perhaps. However Euler did conjecture the functional equation for zeta (in a slightly modified form) where "the gamma function" thus appears together with zeta. So, then the answer to your question is also somehow yes (in a special case).

Not sure what this means regarding acceptable. If you would make the effort to phrase a nicely written question a long the lines you suggest, I would not have anything against it, and develop the above a bit as an answer (except noone else does it first). But perhaps somebody elses opinion would be interesting to have in addition since I might be biased.

Or, you just try your luck and do not get offended if the question is not well-received. (Some time ago a moderator rather suggested that this is in general the better procedure.)

You are welcome. The issue is that at Euler's time analysis was not (always) that rigorous. By todays standards some things that were then considered as a proof, today are (at best) only a heuristic. (To clarify this is a general assertion, I do not know the details in this particular case.)

Added: it occurs to me that there could be another source of confusion of OP. I claimed Euler only considered the questions for integers. Which on closer inspection of the paper however does not seem true. So, yes, Euler knew that gamma and zeta are somehow linked; he did however not prove the functional equation except in special cases.

@voloch: Sorry for the noise! It's true I likely did too many today; I thought since I left some breaks it would be still alright. Somewhat ironically, I had decided to stop for now before your message. But I will keep it in mind for the future.

Yet, if you start looking there are just so many tagging-oddities (e.g., prime was used three times, once for ideals, twice for numbers, both other tags prime-ideals and prime-numbers exist and are more frequent), and mainly I try to limit to things where I hope for a lasting effect in the sense of getting rid of certain unfortunate tags entirely before they become somewhat popular (cf. also my recent post(s) in the relevant sticky thread in case you would like to see what I mean).

Anyway, no more edits from me for a while, and then I will keep the rate a lot lower. Sorry again!

@Teo B: No, moderators cannot actually retag questions without bumping them either. What they can do is perform certain operations (merging and renaming) with entire tags globally, which sometimes can serve as substitute for individual retagging. For example, for 'prime' a global operation was impossible as the tag needed to be split. (Okay, one could have split of the one, and then merged the two others, saving two of three bumps, but then at least one and perhaps both, of the two had no top-level tag nt, which I added and otherwise could not have been added though this is minor, and the entire alternative operation seemed a bit inconvenient.) In general, as documented by my posts in the sticky thread I do try to balance minimizing bumps and convenience (primarily for the moderators and secondarily for me).

The general motivation for most anything bumping the question is to avoid 'evil' edits. If tagging would not create bumps somebody might do 'evil' or perhaps just 'funny' retags without them being noticed.

Yet, indeed, the software tries to be intelligent in noticing what is or is not relevant. To take advantage of this one just needs to view 'hot' and not 'active' also if one were only interested in new questions one could use 'new'. (In 'hot' tag-edits seem to be ignored [evidently also in 'new']; it has however some other disadvantages but depending on ones preferences it could be the better option than the (default) 'active'.)