As a manager, you witness that a male employee harasses one of your female staff but she doesnt complain! Will you take any action?

Yes you should.   Firstly, talk to the female and offer any help she needs.  There are many reasons why somebody who has been harassed might not wish to complain, and that decision must be respected.  Secondly, talk to the male and make it clear that, whatever his intentions and his understanding of the situation, this behaviour is not acceptable.

There is a small possibility that what you observed was not considered harassment by either party.  This situation can arise when the people involved know each other well.

It is important that you act because if he's done it to her, it's only a matter of time before he does it to somebody else. Further, behaviour of this sort tends to escalate in severity if left unchecked.

affirmative! if the female employee does not complain its fine but there is certain conduct that is expected at a work place i will call both of them and ask them to respect that conduct  and leave it at that until the female employee lodges a formal complaint.

i will not take any serious action till any complain, but i personally inquire the relation between them, and search who is crossing the limits and who is breaking the rules then i will take any decision. 

Yes, if it is really harassed situation then i will take action because may be the female employee can not act due to some threats she get from manager or afraid of loosing job or some kind of blackmailing.

I understood from your question that a female employee has been harassed sexually by male employee ,

in this case you have to punish strictly a male employee in spite of a female employee has not Complained from this action.

take an immediet action is not a solution.let them to remember the norms and discipline of the company at first stage

In office this is illegal is he and she like each other than they enjoy in home.

first is to find out what it was all about from both parties in a manner not to suggest any alarm before reacting to it.

A suit doesn't come to effect unless the matter comes to court, but well this can't be avoidable at the workplace where every employee are expected to maintain professional etiquette. Nobody likes to be harassed and the matter has to be put forward to the governing authority before it is worse under repeated occurrence. There may be the circumstance where female staff is obliged not to complain, like she may be victimized for a blackmailing situation or something like that. The Manager, is some governing body who is endowed with the capability to foresee the uncertain future which may be the result of current incident and he cannot avoid this situation. Hence, for a good governance, he must call upon the concerned party and resolve the situation diplomatically. Thank You!

The manager first should take HR into confidence. and separate meetings should be held with both.