Getting into a fight with someone, I think the other person is accusing me of being the wrong one and is trying to show that everything that has happened is my fault. Stop shooting the ball to my opponent is what I would say because it is what I would translate the idiom in my native language into. Does it make sense to native English speakers? What is the common English idiom in such situations?
An English idiom for this is throw under the bus. It means to pass on the blame to someone else (can be friend or foe) who doesn't deserve it, usually out of malice or personal gain.
At least as the question is explained by @MasterPJ, an expression with a similar meaning is "Whose side are you on?" (An emphatic "anyway" at the end is optional.) This does indeed imply that someone who owes you allegiance is being overly generous to the other guys.
"Whose Side Are You On" is a song of the American labor movement; the answer, of course, being the unions' and not the bosses'.
No, this idiom does not make sense to a speaker of English, and I did not understand it when you said it to me. The word you want is "shirk", although I wouldn't really call it an idiom. So you could have said "stop shirking your responsibility", or something like that. Or you could just have said "you need to accept some of the blame for what's happened".
Note that the word "shirk" has another definition that is not mentioned in either of these two references. I believe that the other definition is almost entirely unknown to non-Muslims.
We have a similar expression in English:
We say this to mean, The responsibility lies with you now. We also say, I am putting the ball in your court, meaning, I am making this your responsibility. I suppose, by that reasoining, you could say, Stop putting the ball in your opponent's court. Take some responsibility!
Translation of sentence: "Stop shooting the ball to my opponent" in quite often used in Czech Language and as we can see from others, it is not understandable for English native speakers.
The meaning is (as Mitch commented): A friend/team-mate/colleague accusing you of helping his enemy/opponent/competitor to have an advantage over him.
(I am not a native speaker so I do not know the appropriate English equivalent, the closest I found is Fifth columnist, I hope this post will help to identify better one)
I would say that it doesn't. "Stop shooting yourself in the foot", is a possible English idiom for this type of situation.