When a female says that she has to "powder her nose", we know what she means. What is the male equivalent of that phrase?

There is no male equivalent, men just say "I need to use the bathroom."

Men generally do not put makeup on their faces so it can not be used as a polite excuse.

"I have to go visit the little boys' room" is one, and there are many others.
http://www.usingenglish.com/reference/idioms/powder+your+nose.html
http://thesop.org/index.php?article=14976

"U.S. pissoir (French) Gents or Ladies, little boy's room or little girl's room (informal)"
http://www.thefreedictionary.com/go+to+the+bathroom

A man will say,"I have to see a man about a horse" when he needs to use the restroom.

http://www.urbandictionary.com/define.php?term=see%20a%20man%20about%20a%20horse

I've heard men use the phrase "to powder my head". When a woman says she needs to "powder her nose" she does not mean she needs a toilet, just a sink.

In the UK, to powder one's nose implying a withdrawal to do so is definitely a dash for the loo. If it's actually the nose that needs powdering, they just do it. We haven't powdered heads since Hanoverian times - I've never come across that expression here as real thing or euphemism. And it's 'a dog' rather than a horse here, too. 'Going for a Jimmy Riddle' can be used (but not at the Savoy or Buck House). 'Little boys' room'? You'd be regarded as so twee as to be unbelievable. (They'd assume you said 'Up the wooden hill to Bedfordshire' too.) 'Loo' is quite acceptable in most circles. Only the Hyacinth Buckets would cavil at that. https://en.wikipedia.org/wiki/Hyacinth_Bucket (Incidentally, Bucket is a real surname. A young friend of mine was a Bucket until her parents divorced. Pronounced bucket not bouquet.)