How to reconcile with an old mentor after (unintentional) plagiarism accusations

Nobody is going to mention that the OP's old mentor put undergraduate work directly into a publication (i) without checking that it met professional-level standards of academic honesty and (ii) without informing the undergraduate that this was being done or that he was being added as a coauthor? In my view, these transgressions are equally bad or a bit worse than including a few sentences without quotation marks from sources that have been cited in a survey paper.

In my opinion there is a good chance that the mentor is embarrassed that these lapses of his own have been exposed, which complicates the OP's situation. I am surprised that other answers seem determined to pin this all on the OP. @Scott Seidman suggests that the OP contact the editor:

I think an email to the editor involved, cc'ed to your mentor, taking responsibility (without qualifiers) and stating that none of the other authors were in a position to recognize or correct this prior submission,

In my view the mentor was in the position to recognize and correct academic cultural errors made by his undergraduate mentee, and since the OP was not involved in the preparation or submission of the manuscript in any way, the responsibility for this clearly lies at least partially (and I would argue, primarily) with the mentor. Sending such an email to the editor could just call attention to his poor mentorship and supervision: i.e., it could actually make it worse. (Pro tip: don't contact an editor about something unless you're sure all your coauthors are on board. If there is any real doubt, clear it with them explicitly. If someone is not returning your emails, you can't do this.)

The OP is not blameless in the situation: he made a mistake, and a mistake involving what sounds like a borderline instance of academic dishonesty. He should apologize to his old mentor -- which he has done -- and move on. I would recommend that the OP continue to treat his mentor like an old mentor: i.e., send him occasional emails keeping him up to date on his current academic life. I would certainly not assume that someone never wants to hear from me again based on a single unanswered email. But ultimately the mentor should know that he made mistakes as well, from a position of far more experience and responsibility. It does not fall on the OP to take any kind of heroic or (especially) self-abnegating measures to remedy the situation. One hopes that eventually these two people can move past one negative interaction amid many other good ones.

If you are in the US and you professor is in India, I don't think a viable option is to visit him (at least for now). That leaves you with email and phone. If your professor doesn't answer emails, you could try to call him. However, that might be really awkward, given that he doesn't even send you an angry/disappointed/indifferent email. I don't know how many times you tried to email him, but I would try at about 2-3 times. If you get no response, leave it be, you won't get any benefits from spamming him.

If you get the chance, visit him and try to work it out eye-to-eye. Ultimately, though, I wouldn't go out of my way to reconcile with him at all costs. Some people just don't forget/forgive some slight, regardless if it was intended or not. In my opinion, if you already reached out and explained your "unintentional mistake", that's as far as I would go. Now it is your professor's turn to reply. If he doesn't want to, you can't force him, nor let that fact run your life.

It is hard to know for sure how 'bad' your plagiarism was without personally reviewing it, but from what you explain, I do not think your situation is that bad.

As others have mentioned, professors are busy, and as you emailed once, it could be that he does not have time to reply, is working on something before replying, or just forgot.

If you would like to follow-up, I would suggest apologizing while giving your explanation you gave here. As long as it's honest, and how it sounds to me, you can apologize not for plagarizing only, but more about misunderstanding the situation and the work you left behind.

Explaining it as you have here, something along the lines of (but in your way, I'm not the best at writing these letters, nor understand the cultural nuances you will have to include):

Dear Professor,

I am sorry to find out I have caused problems for you with the literature review I had done. At the time, I had not felt it was completed as there were outstanding issues such as the lack of originality and the heavy reliance on past reviews. I was surprised to find I was being included as a co-author on the submitted manuscript, and realize I should have mentioned the problems with my review at that time. I hope you do not feel I had done this intentionally. If you would like me to redo the review properly in order for it to be submitted again please let me know.

Sincerely,

GKS