The earth will not collapse! Gravity, which is the reason that objects fall towards the ground, is strong enough to hold the planet together, but not strong enough to make it collapse. Though you could say that in a way, the earth already collapsed. A long time ago, it was just a cloud of dust in space, but it "collapsed" in on itself (because of gravity) to form the rocky planet we live on today. However, unlike the earth, stars CAN collapse when they get very old.

For a detailed answer about why the earth doesn't collapse (but stars do), please check the following link to a related ScienceLine question: here.