Minecraft – Can we starve to death

Yes, food now only replenishes your hunger bar, and if the hunger bar is high enough, you will regenerate health at a fixed rate (~1/2 hp/5 sec at 90-100% sated). Conversely, and depending on difficulty, when you're starving you will either lose health until you hit 50% (easy), 5% (normal) or die (hard).

Various actions including walking, swimming, running, and stabbing things will increase your hunger (though notably not boating).

Food now stacks to 64, so it should be plenty easy to carry around enough for quite a while.

If you don't want to die while fighting hostile mobs, don't be so reckless; wear armor and/or take them out at range with a bow. As of 1.9 and 1.0 release, you can also use potions (Potion of Instant Health and Potion of Regeneration) to supplement your longevity. A Golden Apple also gives the same effect as the Potion of Regeneration, though owing to the extreme rarity probably not worth it.

In my experience, sprinting while jumping around depletes the hunger bar more rapidly than anything else I do. This is confirmed by this Exhaustion Level chart taken from the wiki:

NOTE: Any action not listed here will not increase exhaustion level. For example, the player can travel several days by boat and the food bar will not decrease.

Food poisoning seems to work rather effectively as well, so chow down on all the Rotten Flesh and Raw Chicken you can find for additional exhaustion while you sprint and jump around like a madman.