Connect a 115 volt peristaltic pump for acid injection to a 220 volt well pump?

Are you asking if it is possible?
Are you asking if you should do it?

Your post is very cryptic.

It CANNOT be done either way if the well is a straight 240v since there would be no neutral present on the circuit. And this is most likely the case.

There are two 121v(black and white)lead wires and one ground(green)wire connected to the pressure switch. The peristaltic pump has black, white and green. When I check the current from either one single, black or white at the switch and the ground, I get 121v (if both together then is 240v). Can I connect the black from the peristaltic pump to the black on the pressure switch and the white and green wires to the ground at the switch? I've been told that will give me the 115v needed for my perisltc. Pump. But I want to make sure that's OK before I go ahead.

There are two 121v(black and white)lead wires and one ground(green)wire connected to the pressure switch. The peristaltic pump has black, white and green. When I check the current from either one single, black or white at the switch and the ground, I get 121v (if both together then is 240v). Can I connect the black from the peristaltic pump to the black on the pressure switch and the white and green wires to the ground at the switch? I've been told that will give me the 115v needed for my perisltc. pump. But I want to make sure that's OK before I go ahead.
100% absolutely NO. Whoever told you that was very wrong. DO NOT listen to them regarding electrical advice.

Like I said, the well is straight 240v, with NO neutral. You MUST have a neutral to get the 120v you need.
Yes, the well hots will read 120v to ground, but that is meaningless, it is still ONLY a 240v circuit.

OK. I really appreciate your advice! How can I connect this pump in a way that will be tuned on (and pump acid into the irrigation line) when the system (pressure switch/well pump)are turned on? This peristaltic pump was connected to my home's well system (220v 1 phase 3HP pump) before, to inject chlorine to the water line. I know it was connected to the pressure switch but a don't remember exactly how. I do remember both wires black and white were connected to the pressure switch and green to ground. The irrigation pump I want to connect it to now is a 220v 1 phase 5HP pump, the connection box looks more complicated than my house's well, there are extra capacitors and the connection board has a contact labeled (sw), were the black wire from the switch is connected to, while the white is connected to L2.

If that same 120v pump was connected to a 240v well before it was illegal and unsafe.
Most chlorine injectors I see are 240v so this is not an issue.

To do it now you'd need a relay with a 240v coil.

Any thougths on how to connect the relay to the pressure switch and the peristaltic pump? Will any 240v relay do, or does it has to be a specific one?
Thanks.