Brazil is the most important country in the history of mixed martial arts, but not in the history of the UFC.
When the UFC burst onto the scene in 1993 and had a Brazilian by the name of Royce Gracie as its poster boy, the whole concept of "mixed rules" fighting was a new thing to contemporary American sports scene, then obsessed with boxing and unrealistic Karate/Kung Fu films.
However, "no holds barred" fights were nothing new in Brazil. These sport of fighting was called "Vale Tudo" which was Portuguese for "anything goes," and it became immensely popular in Brazil in the 20th century.
The biggest stars of the Vale Tudo scene?
The Gracie family.
Their patented weapon?
Their fighting art (which was basically Judo modified to focus more on ground techniques rather than throws and arbitrary competitions for points) was eventually exported via Rorion Gracie, who traveled to the United States and, after some struggling, managed to get on his feet and even break into Hollywood, where he is famous for teaching Mel Gibson a bit of Jiu-Jitsu.
Rorion's exploits, as well as those of the Gracie family, were written about in an edition of Playboy magazine that caught the attention of an advertising mogul by the name of Art Davie. Davie and Rorion came up with an early version of the UFC (then called War of the Worlds).
This meeting, along with the help of pay-per-view executive Robert Meyrowitz, led to the creation of the UFC and the airing of the inaugural event in which Rorion's relative Royce Gracie showcased how effective the family art could be against someone who didn't know it.
But it wasn't exactly the family art.
What they (brilliantly) branded and marketed as "Gracie Jiu-Jitsu" was actually Brazilian Jiu-Jitsu. The Gracies didn't invent it; they just packaged it and sold it better than anyone else.
Nevertheless, it started a martial arts revolution.
Soon after the first UFC events, serious fighters began training in all facets of Jiu-Jitsu. You had to know BJJ, or at least how to counter it, if you were going to be successful in the cage.
This lead to strikers cross training in BJJ. Once the BJJ fighters realized that the strikers knew what they knew, they started training in striking, and mixed martial arts was born.
Thus, Brazil and BJJ helped to create the modern sport of mixed martial arts, but it wasn't the most country in the UFC's history.
Yes, a Brazilian with a Brazilian fighting style helped create the UFC, but that was the old UFC, the UFC owned by the Semaphore Entertainment Group.
The modern-day UFC is owned by a company known as Zuffa. And it's Zuffa that's responsible for much of the UFC's current success.
Zuffa backed the UFC even when it was making a loss, and eventually got the UFC onto Spike TV in the form of a reality show called The Ultimate Fighter. The show catapulted the UFC into stardom practically overnight; it had a place in society now.
The UFC would continue to grow and hold events in other nations, but its principle fanbase was in the United States, and more UFC shows were held in the United States than any other country (since it was and is an American company, after all).
But that's not to discount what Brazil has done for the modern UFC. Many of the Zuffa era's best fighters—such as Anderson Silva, Junior Dos Santos and Jose Aldo—are all from Brazil. The Brazilian market is also a massive one for the UFC, and the MMA circuit in Brazil is ripe with amazing, undiscovered talent.
Still, while it was Brazil that ultimately created MMA and the old UFC, it was the actions of Americans in the United States that helped bring it to it's current heights.
Both countries have their importance, and modern MMA/the modern UFC couldn't exist without either.