I have resigned from my current organisation as I got a good offer. My manager promised to give me level raise and a counter offer for which he asked me to show the offer letter so that he could discuss with HR team for same. I forwarded him my offer letter and the email that i received from my new organisation. Now, Will the HR team cross check or contact my new organisation to check the authentication of the offer letter? If it happens, will it harm my new job offer in any way?
I don't think they will check, and if they do, that is not a real problem, you are free to resign from a position while explaining you have a better offer, then the company might give you a counter offer, I don't see any problem here.
If it happens, which I doubt, from the point of view of the other company, that will add to your value, you will be seen as someone who is enough valuable to do efforts to keep it.
The only bad side may be that the new company might see the leak of the offer as a intent from your part to play a double game, which would be problematic for you. They will think you are not trustworthy. It really depends of your country.
Accepting the counter offer is also really dangerous. Because it will put you in a strange position with your colleagues and your hierarchy, adding to the fact that they will think about you as "the guy who is living for money".
In your position, that is a game I would have not played.
It's highly unprofessional to ask the HR for the offer letter. It's worse you send it to them. And it's worst if they call the company to check its correctness.
I can almost guarantee that that's not gonna happen though: I can't even imagine they pick up the phone and ask "Hey, would you confirm you are actually offering this, so we will continue bargaining with him over this?"
For future, you don't even need to mention about the other offer.