No. There were several other VA's in the cartoons.
However, between about 1943 and 1960, Mel was the only VA to recieve screen credit, and prior to that NO VA's recieved screen credit at all, NOT even Mel!
In addition to Elmer Fudd, Arthur Q Bryan originated the Fudd voice before Fudd even materialised, in Tex Avery's Dangerous Dan McFoo (1939), voicing a pup character with the same Fudd voice. In Nutty News (1942), Bryan's ONLY Black-and-white LT, he used the Fudd voice as an announcer. And in A Pest in the House (1947), Bryan voices the tired businessman (whose voice sounds VERY close to Bryan's natural voice) in addition to hotel manager Elmer Fudd. When Bryan passed away in 1959, Mel was originally asked to replace him, but Mel was very reluctant to take the role, so the role was taken over by several other actors, namely Hal Smith and Dave Barry, but later Mel gave in and voiced Fudd until he died.
Stan Freberg is worth noting as well. He started voice work for WB at age 18, and some of his roles include Pete Puma from Rabbit's Kin (1951), Junior Bear in Chuck Jones' "3 Bears" cartoons, and practically ALL the voices and the narrator in The 3 Little Bops (1956), the only time he recieved screen credit, and Bertie in Chuck Jones' "Hubie & Bertie" cartoons, among other roles.
Kent Rogers had a very short-lived career, as he was tragically killed during WWII, but he was famous for voicing Beaky Buzzard in Clampett's Bugs Bunny Gets the Boid (1942) and The Bashful Buzzard (1945). In addition, he voiced a scientist whose voice was a takeoff on British character actor Richard Hadyn (who most animation buffs will recognise as the voice of the catterpillar off of Disney's Alice in Wonderland) in Chuck Jones' Super Rabbit (1943).
Billy Bletcher, who is famous as the voices of Pegleg Pete and the Big Bad Wolf at the Disney studio, voiced Steve Brody in Arthur Davis' Bowery Bugs (1948) and Papa Bear on the Chuck Jones "3 Bears" cartoons (except Bugs Bunny and the 3 Bears, in which the papa bear's voice is provided by Blanc).
Pinto Colvig, Disney's Goofy, has also done a few voices for WB as well. He voiced a Kangaroo in Norm McCabe's Hop and Go (1943), and voiced Conrad Cat in Chuck Jones' Conrad the Sailor (1942). To this day, I wonder whether they will hire Bill Farmer to do these characters' voices if they are revived in a "modern-day" LT production.
Robert C Bruce has done a "serious" narrator-type voice that you may occasionally hear in some cartoons, especially "travellogue" cartoons.
Can you recall Sniffles mouse? His voice was done by (I believe) Bernice Hansen, a radio actress who specialised in high, squeaky voices.
I'll let others add to this explanation.