I've done a few approaches at the gym and it is an environment that can turn things really ugly if you aren't careful. Something you need to consider is that you are going to be running into the same people on at least a semi regular basis. There is nothing wrong with approaching and it can be done, but it creates a set of circumstances that can be awkward for both parties very easily. If you are a man and get rejected or she seems to be flakey, immediately abort and don't talk to her ever again. From my personal experience, too many woman today are addicted to attention and will think nothing of leading on some guy at the gym just for some attention. It's not at all normal behavior but it is something that is quite prevalent and I don't think a lot of men are aware of it.
A few years ago I approached this girl at the gym that I had started seeing there and we spoke for about 10 minutes. I saw her a couple days later and she approached ME, we talked about what we liked to do and she more or less asked me out to go hiking, so naturally I accepted. She flaked on our date, to which I immediately had a really bad gut feeling about, and then ghosted me for a week. She then used me as an orbiter that she led on, repeating the same behavior. I ended up confronting her and asked her if she had a problem with me and she finally left me alone but then months later tried to approach me, I literally walked away. I later found out she did this to tons of other men regularly a completely lied about what she had done in order to make me look bad.
This story above is a more extreme example but people need to be aware that this does happen, my other approaches all went went well initially but resulted in something similar.