Regarding doubles match: On our opponent's second serve to me, I called it out. Our opponents contested my call. My partner said she did not see exactly where the ball bounced. Our opponents said because my partner could not confirm my call, it was their point. I said that unless she was clearly able to see that my call was wrong, my call stands and it was our point. Who is correct?
This is more of a comment but too long...
I used to help run tennis tournies in my teens. I made people went to the right court, had balls, had towels, cleaned up trash, whatever.
I also helped deal with disputes. First almost no tennis tournaments have umpires in the US at least - unless there are school affiliations, meaning it is a high school or college tournament.
Second, all tournaments have bylaws on calls. It is very normal for the person on to make the calls on their side of the court. So in your case the person receiving serve would call fault on outs.
Third, there is a mechanism for disputes. Meaning that the other team would be able to argue that the ball was in. Normally in your incident you call a fault, the other team argues, it is simply a redo. And the redo would start at first serve with any tennis tournament rules I have followed.
Fourth, there is a mechanism for too many disputes. So in tournaments I have worked in after the third dispute then someone like me would "watch over" the game. If I believed that someone made a blatantly bad call (yes just one) I had rights to call game as forfeit. If there was a close "bad" call I would warn the person and based on tourny rules or member standing or whatever someone might get 1-2 warnings then forfeit. The idea is if a worker had to babysit and then there were still issues that the issues probably had a dramatic effect before I came over.
And the tennis community is rough - at least in my area of US. If you were in a tourny that used NTRP rating and got kicked out or forfeited a match you may have a hard time playing in a tournament anytime soon as it is too costly for clubs/event places to hire umpires for every match and they simply don't want to deal with these types. We had a guy that had to petition to get back into tournaments and write an apology letter as he was banned for three years (very funny, he was kicked out of next tourny in his second match).
So to answer your question - NORMALLY - in friendly tournaments without umpires which is probably normal in the US... you call out. Either your call stands or opponent disputes (if in tourny rules) and the entire point starts over. There would simply be no tournament laws that would say your opponent says it's in so it's in. On the other side if you had been making sketchy calls all day your teammate would maybe be trying to save you - not saying this is the case.
So to sum it up in any tourny or even competitive friendly neither of you were correct. The point should have been played over if that was your first dispute of the game.
Note - we did try one tourny for the non-receiving doubles partner to make the in/out calls for serves and this last like 1 hour into the tournament. As competition gets better the other player is at the net focused on the movement of the other team after the serve not looking back at their teammate.
The rules of tennis specify that the umpire decides decides whether the ball is in or out (assisted by line judges etc as appropriate). As you were apparently playing without an umpire, you were not playing by the official rules, so it is impossible to say who was "correct".
I suggest that you agree this sort of thing with your opponents before starting a match in future to avoid this kind of issue.