Word a male can use to refer to a female that is not old or young (girl/lady/woman)

I'm a married male in my mid-thirties and meet a woman of a similar age and talk about business.

I want to refer to this meeting later, and it feels weird to say "I met a girl today...," (feels like it describes someone younger or it was romantic) or "I met a lady today..." (feels like it describes someone older.)

Is the best term just "I met a woman today..."?

Depending on context, all three could be acceptable, with woman probably being the most neutral.

However, in all three cases, you seem to want to put emphasis on the fact she was a woman... if you met to talk about business, you could use a job-title.

You could refer to her (depending on the nature of your business relation and her job description) as a colleague, a designer, a manager, a developer, an investor, an analyst, etc.

Apart from professional relationships, you could simply go with someone, a person or an individual, as Qaz notes in the comment.

I think the most neutral way to convey that you spoke to someone and she happened to be female would be something along the lines of:

I spoke to this analyst today, she gave me some very good insight in the business processes.

If you really want to emphasize that she was a notable person, I would suggest to go with woman:

I met this great woman today! Her insight in the business processes was amazing!

Although lady could be used as well - it can convey respect (for instance respect for her professional capabilities), rather than just age:

That lady I met today had some really good ideas about improving the business processes!

I like this question, because I have often felt the same way.

The reason is that there is no word for a female that is quite like the word "guy". "Gal" is often paired with "guy" but "gal" is like a cowgirl or a country girl and is extremely informal, bordering on demeaning. On the other hand, you can use "guy" to refer to any male, any age, any position, without even a hint of connotation.

"I met a guy today..."

I think you have to go with "woman", even in circumstances where you would never say "man". Even though it's true that not mentioning gender circumvents the problem, it's a flaw of current usage that this option exists only for males. Maybe we should all start using the word "gal" and drive the awkwardness out of it!

Being of female gender, I personally would prefer the designation suggested by the person who proposed, "I met a potential business contact today. She was around my age, and had keen thoughts on the ......." This conveys the important elements of the exchange, as in "...potential business contact..." and then adds an approximate age identifier, as in, "She was around my age..." and then adds an important element to identify what the speaker defines as her value to the business being discussed, "...(She) had keen thoughts on ....." This approach is clear, concise, and respectful.

Seems ok to just use "someone" or the job title: "I've met somebody from their marketing department yesterday, she's completely incompetent and bullies her assistant". To the use of "guy": I more and more often hear girls/young women refer to each other as "guys" - "What you guys wanna drink?", etc. Could it be possible that in the near future "guy" becomes an asexual term?

To me, it says a lot that, when you explained the scenario to us, you used the word "woman", and not "girl" or "lady".

This strongly suggests to me that, coming from you, the word "woman" is more natural and less stilted that either of the other two choices. That's what I would go with.

As far as watching one's language to avoid giving offense, there are times when being tactful is helpful. On the other hand, trying to win the favor of people who make it a point to be habitually offended is a thankless exercise. This is one of those times when your own natural usage is the best guide.