Is Ulfric Stormcloak capable of a Dragon Shout?

Ulfric is capable of shouting because he studied with the Greybeards but couldn't stand being away from the events of the war and as such left to join the Empire. He is not Dragonborn though: I think the idea is that almost anyone can learn a shout if they are super dedicated (since it takes a lot of time for non-Dovahkiin).

He also states that he Fus'd him to the floor and finished him off with a sword.

And no, there isn't a shout available to the Dovahkiin in-game that is powerful enough to disintegrate an enemy. That's not to say there isn't a possibility of one in the world of Skyrim, just that you can't learn it in the game.

Just for reference you can find some more information about Ulfric's past here.

EDIT: The Dragonborn DLC allows you to get a passive power that adds a chance for Unrelenting Force to disintegrate enemies. Thanks 3ventic! :)

All 3 of your questions are answered in dialogue with Ulfric himself. If you go see him before you go to High Hrothgar and before asking him about joining the Stormcloaks, you can ask him about the Greybeards, Dragonborn status, etc.

He tells you that he was chosen as a lad by Arngeir to become a Greybeard but he couldn't dedicate himself to the Way of the Voice that required detachment from people and events in Skyrim. He and Arngeir are estranged over his departure. He studied approx. 10-12 years to learn the 3 words of power for Unrelenting Force. This is the only shout he knows, but even the Masters of the Voice in ancient times before Jurgen Windcaller and the Greybeards (see Forelhorst Journal) only knew the Force shout. All Deathlord Draugr (highlords of the Dragon Cults) know Unrelenting Force.

There is no known shout to disentigrate foes. They can be weakened by "Marked for Death." Theoretically, the dragon word for "disentigrate" would do just that but it is not in game (assuming dragons have a word for that in their language).

Ulfric also relates the Duel with the High King. He is aware of the exaggerated tales of his use of the Voice and admits using Unrelenting Force to throw Torygg "but it was my sword that pierced his heart."

To add to the other answers (which are mostly lore-based), gameplay-wise, the Ulfric Stormcloak NPC is capable of a dragon shout.

You can notice Ulfric Stormcloak using Unrelenting Force (Fus Ro Dah) if you fight him in the Battle for Windhelm (part of the Imperial Legion Civil War questline) or if you fight beside him in the Battle for Solitude (part of the Stormcloak Civil War questline).