Obviously there are some, very minor differences between the two men.
For example, de Gaulle was a bit of a stubborn authoritarian egomaniac, hellbent on having things done his way, and when things didn't go his way, he'd take his ball and go home in petulant-child-like fashion. You would never hear words like that uttered anywhere near Gingrich.
Seriously, don't utter words like that anywhere near him.

The most obvious parallel between the two is the military experience. De Gaulle was a war hero who was wounded and captured by the Germans during World War I. In World War II, before becoming the leader of Free France, de Gaulle commanded a division against the Germans.
Gingrich, meanwhile, while having no actual military experience whatever, including in the great conflict of his generation, the Vietnam war, is leading the charge in the next great conflict:

"This is World War III," Gingrich said. And once that's accepted, he said calls for restraint would fall away...
There is a public relations value, too. Gingrich said that public opinion can change "the minute you use the language" of World War III. The message then, he said, is "'OK, if we're in the third world war, which side do you think should win?"

At a later stage in World War II, de Gaulle would become the leader of Free France and play an instrumental part in the liberation of France.
Gingrich, in turn, was a revolutionary in the great American revolution. Okay, the second great American revolution; the Republican revolution of 1994. Following Gingrich's ascension to the speakership in 1995, Gingrich was such an effective leader that his fall from grace and forced resignation occurred less than four years later, in 1998.

But if we had to pick just one thing the two had in common, it would have to be family values. Values that both men spent their entire lives fighting for. De Gaulle married his Yvonne in 1921, a marriage that would last until his death in 1970. He is of course also remembered for his devotion to Anne, his beloved daughter, who suffered from Down syndrome and tragically died at the age of 20.
Gingrich is quite the devoted husband himself, having been married for close to 40 years. Though to three different women. But if you disregard the multiple extra-marital affairs, one of which occurred at the height of the Bill Clinton impeachment investigation, he's been entirely faithful to every single one of his three wives.

Oh, and both their last names starts with "G".

And should the parallels between the two continue, we can expect Gingrich to come back from his political exodus and return to power in a quasi-coup-like fashion. As Gingrich himself says:

I am not 'running' for president. I am seeking to create a movement to win the future by offering a series of solutions so compelling that if the American people say I have to be president, it will happen.

With Général Gingrich at the helm, I'm confident we'll all have "une certaine idée des États-Unis".