No, your provider can't give you a recording of your phone call, and it's not saved on the phone by default.
There are a zillion apps on Google Play for recording phone calls, but there are two things to be aware of:-
Some phones simply can't record calls. This is because the "cellphone chip" in your phone has direct access to the microphone and speaker, so the phone call sounds don't ever pass through Android's phone system. If your phone was designed this way, probably you can record your side of the call (because Android still has access to the microphone), but not the other side. It doesn't matter which call recording app you use, because it's a limitation of the hardware.
In some countries, it's illegal to record a phone call without the consent of the other participant(s). Before doing anything, you should check, especially if you intend to use the recording as evidence to confront the other participant later or in a legal dispute.