Yes, it's possible to close the holes permanently and without needing to get a surgery. However there are three possible situations, and you must check up the hole for properly identifying.
In all the procedures it's necessary to sterilize the area before with alcohol 70.
1 - "Fresh" holes - this is a not healed hole, where the tissue inside isn't taken over by skin. Some people will still have fresh holes after some months.
To close those ones permanently, just clean up the inside with a alcohol sterilized sewing needle by passing the needle eye spinning through it.
This process will remove dirt (that will get inside the needle eye) and allow proper healing.
2 - Healed hole.
This kind of hole have a layer of external skin covering partially or completely the inside of the hole.
This way, the hole isn't going to close unless this skin is removed.
Best way to accomplish this? With a syringe, saline solution and small wound closing strips...
Get a syringe with a injection needle that is larger than the hole, drill with the needle through it (this WILL cause a bleeding when done properly), then use the needle to remove the skin in the hole's endings, use the syringe to pump saline serum through it and apply the wound closing strips on the earlobe for forcing the hole to close. Keep changing the wound closing strips for 5 days. After one week you might need to remove more skin in the hole's endings and repeat the process if it wasn't enough (the hole inside however will be completely sealed at this point).
IMPORTANT OBSERVATION: You might need another person to perform the procedure for you if you don't have a pair of enlarger mirrors or another device that allow you to view your own head side's.
3 - Enlarged healed hole.
Basically this is a hole similar to the healed one, however it got enlarged by the use of heavy jewelry to the point that a injection needle isn't large enough to cut the hole's inner walls.
In this case?
Do the same process above, but replacing the injection needle with a small sterilized sharp drill bit that is just a little larger than the hole. The best drill bits are the very sharp ones used in wood or steel.
Those procedures are used by some body modders and is your responsibility to use this knowledge properly and be aware of any infection signs.
In case of infection, go immediately check it with a physician.