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: 
Bullies enjoy power and hate it when that power is worked around or removed.
Best way for that to happen for you to get alternative proof of employment and any proof you can of the situation you are in.

1. Get copies of any e-mails relating to the complaint. Anything that goes back and forth to show that it is an on-going complaint.
2. Send an email to the director directly asking why he won't provide/is withholding such a letter or in some other manner to get it documented.
3. Take photos (observe company and data security) of you clearly at work and possibly the position/work you to confirm role.
4. Any other documentation to show your working there, contact, wage/pay slips etc.

Take this to your new employer and show that your old director is stone-walling you in an attempt to bully you in to dropping the compliant and that this is why you cannot get the letter required.
See if they are able to drop this requirement given the issues that you are facing.

If this works then finally:
5. Raise a second complaint against this director for his actions and pursue that as well.
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.