You can't apologize for making someone feel badly about something if you still hold that something to be true.
You can express empathy with the other person though. And in English, at least, this is commonly done using apologetic language, "I am sorry that you feel that way and I understand that you are upset" etc.
It is possible to totally disagree with someone but still feel empathy for their reaction to your opinion. As such, you certainly can "apologize" for making someone feel bad even if you don't find the cause justifies it. You're just not apologizing for your opinion.
You can express empathy with the other person though. And in English, at least, this is commonly done using apologetic language, "I am sorry that you feel that way and I understand that you are upset" etc.
It is possible to totally disagree with someone but still feel empathy for their reaction to your opinion. As such, you certainly can "apologize" for making someone feel bad even if you don't find the cause justifies it. You're just not apologizing for your opinion.