No, that's not true. Iceland was Iceland long before Europeans settled Greenland. It's also not (strictly) true that Iceland is green and Greenland icy as absolutes. Iceland can get very cold and icy and snow-filled during winter and has some impressive glaciers. The coasts of Greenland often are quite green and pleasant.
Iceland's name is typically attributed to Hrafna-flóki (raven-flóki, the first settler to intentionally sail to Iceland with the intent to settle around the 9th century. While he probably had some high hopes for the new land his first winter was nothing short of a disaster, and he set sail back to Norway in springtime.
Whilst waiting for spring he walked up to a mountain and looked in to a valley that was filled with drift ice. From there he named the entire island Iceland and spoke ill of it to whomever cared to listen in Norway. He'd make a return later and permanently settle, but he didn't really have much opinion of the frozen rock in the middle of nowhere at first.
Greenland on the other hand is named by Eiríkur Rauði, Erik the red. His naming is commonly closer to being "misleading marketing", as he did (allegedly) name it to make it more appealing to settle or to spite the Icelanders who had exiled him on account of murder. However it is just as likely that he just happened to settle in a valley that was green and fertile that year. He also settled quite far south, where it is more mildly weathered. Additionally back then the climate as a whole was warmer, so it's not unlikely it actually was quite green where he settled.
So there you have it: Iceland is Iceland because a disgruntled settler saw a lot of ice, and Greenland is Greenland because of mild marketing and that it was quite green!
Vísindavefurinn on Hrafna-Flóki, with additional native sources at the bottom