Question: I'm working in a software company in India where I have lodged a complaint to the owner regarding a director I have had difficulties with. 

Now I am applying for a position in a new company but the director (whom I complained about) is withholding providing verification of my employment unless I withdraw my complaint.

What would be the best way to proceed?

Answer: 
This is an unpleasant situation - it's certainly not fair or right that the director is holding your verification to ransom in this manner.

That said however - if there is no-one else in the company who can provide you with the verification (Company owner maybe?) then ultimately you're going to have to choose whether this is a stand you are willing to risk missing out on the new opportunity over.

To be clear I'm not saying what the Director is doing is in any way okay or acceptable - but you have to be pragmatic and decide what is ultimately going to be best for you and for your life. 

Good luck!
For some reason I'd make the point to this director and raise another complaint against him about withholding the letter of employment and his attempt of blackmail... I'd bet the owner would love to hear about this too; and bullies will only stop when someone doesn't back down.

I'd also go around him and ask the HR team for a printed letter of employment - and stand next to them while they do it so that there's no chance of it getting intercepted. I'd expect them to be willing to do this as I'd wager that it's illegal to not provide one (though I'm certainly not an expert in Indian law!).

I can't believe that they'd have been told they're not allowed to give you the letter; and even if they have been, if you're friendly with them then I'm sure it'll go fine.