No, an Earth Elemental can't extend its Earth Glide ability to affect any other objects or creatures. (@Rubiksmoose's answer explains why.)
An Earth Elemental might still achieve a similar goal by brute force, for small objects and soft ground.
If an Earth elemental is holding a small object or creature while entering or moving through the ground, they are effectively pushing that object through the earth the normal way. (This would obviously take much more effort and slow them down, so you'd have to decide how much depending on the object and the soil conditions.)
An Earth Elemental is strong enough to drag/push a tiny metal ball bearing (for example) through loose earth or soil, boring a tunnel as the ball pushes earth out of the way. The object doesn't glide, but it's not made of earth so the Elemental can push on it while gliding.
This seems pretty reasonable, with Str 20 and Large size, it should be within the realm of possibility to brute force an object through the ground, unless the ground is stone or dried clay. e.g. something earthworms can't tunnel through would probably need to be smashed (maybe using the object as a hammer).
For objects not as strong as a ball bearing, this will be very damaging to the object! (As a combat move on a grappled creature, this seems potentially OP, especially burying them alive by stuffing them into the ground. But many Tiny creatures aren't quite as small as a ball bearing or marble, or shaped in a way that would move dirt out of the way. So you might just rule the attempt as a Slam attack unless the ground is particularly loose soil that would make it easy to bury a Tiny creature in, if your fist had no resistance, just the creature Grappled in it.)
Maybe also / instead damaging to the Elemental for a strong and hard object like a ball bearing that could work as a rock hammer in reverse; it might be like trying to drive a nail into wood with your hand. But us puny humans don't have skin like Earth Elementals.
(The elemental might have a problem with a small stone: once it takes the stone underground, the stone could be considered part of the ground and the elemental might not be able to glide through ground while also pushing on the stone object.)