What your thinking of was part of the "bowling etiquette" but not a set rule.

If two players arrived at the approach at the same time, the player on the right goes first. Otherwise whoever arrives first goes first.

Also if two players are at the line and one has a spare, the player with the spare  goes first.

Since the player on the left went first, the player on the right I think has the option to wait and allow the other player to finish the frame.

It's kind of a nit picky thing to worry about, I assume somebody is thinking "well if he hadn't waited things would have been different".

As has been stated, I'm 99.9% sure there's no rule mandating who has to bowl and when in these situations. Usually, if a bowler wants to wait so that he can see what he'll need, he stays seated or stands back someplace away from the approach and waits for everything to shake out before stepping up to the ball return. I anchor for my teams, and if I'm determined to wait, I'll usually mark down the scores. That way I'm doing something productive in the meantime. Now, the way you described it makes it sound like this other guy walked up to the ball return as if he was ready to go but then refused to do so until the other anchor bowler had finished the frame. Is that how it happened? If so, the guy who did the watching and waiting made an awkward situation out of something that didn't have to be.

Personally, I'd like to go first and (maybe) get all three strikes.  Wanna see if he'll choke on his "strategy".

Maybe the guy wanted to put himself in a pressure situation. Maybe he likes performing under pressure, to get what he needs. There's no rule against waiting. It's not polite, but it's not against the rules. Also, is that any different than the guy that gets on his cell phone, wanders off, and their teammates are busy looking for them?

Again, they're not nice, but head games aren't illegal. If they were, running shots out, slapping your hands, raising your voice, etc etc would be illegal. Don't let someone get under your helmet, and you will be fine.

Of course there's nothing wrong with wanting to wait and see what you need to do. I do that from time to time as anchor man, but, again, I stay seated or back away from the approach. I don't go up to the ball return and just keep telling the other guy, "no, you shoot again."

From the way the OP described it, it kind of sound like that's what this guy was doing. To me, that's making things awkward and/or taking gamesmanship a bit far for league. If you want to wait and see what you need, stay in your seat or hang back away from the approach. That way the other guy isn't wondering whether or not you're ready to go.

Now, I've even seen this go too far the other way as well. I've seen a guy on one team take his shot and then sit down and refuse to bowl until the anchor from the other team comes up to throw his first one. I think that's stupid, too.

I like to win as much as anyone, but I'm not going to go out of my way to "get in the head" of one of my opponents during league. If we were bowling for thousands of dollars, maybe. But as it is, I'll just take my shots, do my best, and hope that I help my team come out on top. Some guys over-think these things big time!

The head games concept just isn't for me, but that doesn't mean it's wrong. There are certainly some people I enjoy beating more than others. Still, I think it stings just as much when I do it with a smile as opposed to doing it with a scream and crotch chop!