Did the conversation/discussion just end by saying that you have not shown any eagerness, or was there any explanation for that? I am sure if he had not given any explanation, you wouldn't have left it like that. You must have definitely asked for it. What was it?
If everybody else is confirmed, why are you exceptional? Is there anything that is untold?
Finally, if you are not satisfied, then you can leave the company, but I think introspection is definitely needed here. You need to understand what is falling short and work on that rather than switching jobs.
Firstly, I was alone there along with my reporting HR, and as I said, he manipulated the topic towards trivial issues such as "I've allowed a lot of convenience to you, etc."
Secondly, I am not satisfied because why did he not inform me three months back when I was supposed to receive the letter? Why now, and only after when I asked for it?
Thirdly, leaving the company is like giving up what is my right. I deserve the letter and appraisal, but since he has no logical answer to that, he is just deferring. I've seen him doing this with the rest of the employees, but when they object, he sooner or later delivers what ought to be.
I posted this issue here so that I could get quick and competent advice, which I must say I haven't seen yet. Please, if anyone could help rather than investigating. I have already given sufficient information.