No. In fact, he's seen around the last twenty minutes of the film, asking Private Fife (Adrien Brody), "Fife, Fife...where's Witt?" This is when Pvt. Robert Witt (Jim Caviezel) uses himself as bait to draw the Japanese soldiers away and buy Witt some time so that he can run back and inform the rest of the unit that they're advancing closer. As the movie draws to a close, Pvt. Bell doesn't appear again onscreen, however the movie did not show that he died.
Watching The Thin Red Line on Netflix. :)