The question at this point (with a good answer already accepted) is:
Even if you "give it time"(which you should) will you be able to salvage this relationship? And if you continue to text apologies that make it worse, or if you try to talk to her (while mentioning her sex life) at work, will you end up having a chat with HR?
Your problem SEEMS to be that you have feelings that you are not willing to acknowledge, and without doing so you'll continue to give half-apologies or even make the same mistakes again.
@henning had two comments that are along the same lines as my thinking here, and he put it better than I can:
...suggesting that I might be looking for similar attention. This was not my intent. What was your intent?
I didn't want this comment to sound snarky. What I'm suggesting is that OP investigates his motivations, in that feeling lonely and in strong need of emotional support easily leads to (secret) jealousy, insecurity, difficulty to accept boundaries, romantic projections. This is just natural and not necessarily bad; but all kinds of awkward situations might result if these issues are not acknowledged but disowned.
With all the context you've given us, either
A) You have romantic or conflicted feelings about this person
or
B) You may have come across this way, to at least some of us and maybe to her.
Given whichever of these is true, you should approach further interactions with that in mind. If you do not, it isn't just your friendship that will be in jeopardy, your job and your marriage (or half of your belongings) will be in jeopardy.
I know I'm reading into things here, but from everything you've said I can't help but read into it:
I am a married man dealing with some personal issues
Unwritten subtext here is that being a married man is one of these personal issues. If that isn't it, then you'd at least have a partner who could help resolve your "foremost" issue:
foremost among them is a lack of a proper support network of friends
to talk to about the stress in my life. I have been trying to
proactively get involved with new people through work so that I may
eventually develop the kind of friendship(s) I need.
It seems from this and from the next thing you've said that you've talked to your friend about the stress in your life (something some guys do when they're stuck in a bad marriage and trying to find a new relationship), and that this stress may be marital-related:
Foremost among those I have begun to develop some comfort with regarding person topics is a female coworker.
Has this just happened to be the case, or is this "foremost" for a reason?
I noticed a conspicuous simultaneous absence of her and another male coworker. After she left, I made the regrettable choice to send a text asking if something was going on there, with a "good for you" vibe.
Send all the vibes you want, there are very few people who are going to believe that a man texted a woman asking if she's having sex with another man for a benign reason.
My immediate apology included the phrase "I only wished to let you know I am okay with whatever".
You don't really say much more about this immediate apology, so maybe you didn't think it was that bad. From an outside opinion (at least this outsider's opinion) this rings entirely false. If you were "okay with whatever" you wouldn't have gotten involved in something that is so not your business. Part of getting a handle on interpersonal relationships is not attempting to make things look better by pretending at something other than the truth. If you are hiding the truth from yourself, you will be the last one to see it.
What makes it particularly horrific is I did not realize that male coworker was married. My choice of phrasing things has thus taken on a much darker meaning, suggesting that I might be looking for similar attention. This was not my intent.
We all agree that this isn't good, but what makes what you did truly "horrific" is that a man having personal issues texted a woman (that he isn't romantically involved with but who he is paying a lot of attention to) asking if she's having sex with another man... WHILE she may have been having sex with another man (you leave this unsaid in your answer). I wouldn't bring up the fact that the guy was married in any future conversation/apology - apologize for what you did, not something you didn't know.
I left a follow-up apology by text the next day when I realized this. I explained that I would never wish to imply she was playing with married men.
Yes, this did make things much worse. It's an apology for the wrong thing, and it implies WORSE things. She could honestly be wondering if you sent something like this out of jealousy, and if this is a fake apology implying worse things as a result of said jealousy. At this point you need to not only worry about whether she's insulted, she may even be feeling some level of threat, and you have to approach this more carefully as a result.
I am a social idiot
I'm a social idiot too, and I've done things on this level of carnage, and they were motivated by feelings I wasn't admitting to at the time. Understanding that you may have had some jealousy, or that you at least may have appeared to have some jealousy, will help you become less of a social idiot to your friend going forward.
Should I attempt to talk to her (at work!)
NO.
If you give your relationship with this person a higher priority than your job, can you really say you only have friendly feelings toward her? Maybe so, but you are seriously proposing here in front of all of us that you discuss a woman's sex life with her at work (if your apology takes the same course as your previous ones). This is where you stop and go out of town and clear your head and come back and be professional at work. If you have to say anything at work, now or 3 months from now at a birthday celebration when you're stuck standing next to her, make it short and simple and professional, along the lines of: "I want to profusely apologize for being an idiot." But the only real place you can talk about this in depth (which may be a bad idea) is a social gathering outside of work.
I haven't said nice things here, and I've gone beyond your original question of "Do I give this time?" I may start my time here at IPS with negative rep. But if you don't come to an understanding of WHY what you did was wrong, if you continue to apologize for the wrong things by saying worse things, if you proceed with someone that you either have feelings for or who may think you have feelings for them without approaching the problem from that angle, you are not only going to destroy your relationship... you are going to have a problem with HR. And problems with HR can always make it back to your wife.
So, yes. Give this time, and quit texting her.
