Not even close. Although teeth submerged in a cup of soda do eventually dissolve because of the acid present, this process takes several days.
This myth likely originated from a 1950 Congressional testimony by a Cornell University professor named Clive McCay. He was trying to convince a committee of House representatives that soda could cause cavities, and he claimed teeth submerged in Coke would soften and begin to dissolve in two days’ time. His claims, though, were a bit exaggerated, and other scientists noted that teeth would eventually dissolve in any liquid containing sugar and phosphoric acid. Like orange juice.
If the idea of soda dissolving your teeth worries you, keep in mind that no one keeps soda in their mouth long enough to do that type of damage. While soda isn't necessarily good for your teeth, saliva and proper dental hygiene help mitigate the effects of the acid.