Is “Call for somebody’s head on a plate” a popular idiom? What is its origin?

Your speculation is correct. The phrase is from the Bible story of the beheading of John the Baptist by Herod in Matthew 14:

But when Herod's birthday came, the daughter of Herodias danced before the company and pleased Herod, so that he promised with an oath to give her whatever she might ask. Prompted by her mother, she said, “Give me the head of John the Baptist here on a platter.”

(According to this site, Salome is an opera by Richard Strauss, and was based on a play written by Oscar Wilde about the death of John the Baptist. The daughter of Herodias is not named in the Bible; information about her comes from the historian Flavius Josephus, who apparently named her.)

Asking for the head of one’s enemy to be presented on a platter has become a phrase denoting a desire for revenge on an enemy.

An Ngram of head on a plate vs head on a platter shows the latter phrase to be much more prevalent. As for either form being a popular idiom, because of its biblical origin, it's a phrase that many people in the US (and perhaps other English-speaking countries) would have no trouble understanding.

It is a common usage - from the biblical reference that @Gnawme mentioned - and usually means that a person is not going to take revenge themselves, but feels that others should deal with this person. It is often used to say "this person has clearly broken xxx rules, so yyy should probably act"

In the case highlighted, it was used to indicate that Newt was expecting Clinton to be impeached and brought down. It has an implication of wanting to see someone destroyed, while not intending to actually do anything illegitimate to make it happen.