Does a creature with indestructible die from being dealt damage equal to its toughness?

When a creature has damage marked on it equal to or greater than its toughness, state-based actions check whether it should be destroyed for lethal damage (CR 704.5g). Normally, this destroys the creature. However, indestructible permanents cannot be destroyed by any means — neither by the 'destroy' keyword nor by lethal damage (CR 702.12b).

Because the state-based action specifically uses the word 'destroy,' and indestructible shields the permanent from destruction, the creature simply remains on the battlefield with the damage marked on it. The damage does not disappear, but it also does not finish the creature off.

Indestructible does not protect a creature from everything, however. Effects that reduce toughness to zero or less will still cause the creature to cease to exist via a different state-based action (CR 704.5f), and exile-based removal bypasses indestructible entirely, since neither involves 'destroy.'

Concrete example: Your opponent attacks with a 4/4 creature that has indestructible. You block with a 5/5 that deals 4 damage to it. At end of combat, state-based actions see the 4/4 has 4 damage marked on it — lethal damage — but because it is indestructible, it is not destroyed and stays on the battlefield. Next turn, the damage falls off at the cleanup step (CR 514.2).