Can the Earth grow?

Well, it does a bit (farmerman, please confirm). We're hit by meteors all the time (a lot of them are teeny tiny), and they stick because of gravity. Hell, that's how the earth (in some ways; it was also due to particles colliding from the sun's accretion disc) got formed in the first place.

If it stays away from sweets and fatty foods, there's still hope.

It all depends on what you mean by earth. When the earth warms the atmosphere expands which clearly means the volume occupied by the earth and it's atmosphere is more. That doesn't mean the earth has more mass however.