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!
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.