"The contractor put white grout under the first run of the tile where it meets the granite counter."

No grout will stay in this joint.
It should be caulked, not grouted.

Is there a special kind of caulk for using on tile and granite?

Acrylic latex, 'Siliconized' acrylic, or silicone, just NOT butyl.

Thanks

Also, grout companies offer sanded caulks that match the grout color and texture for just this purpose.

I have existing granite countertops with backsplash of the the same granite. The joint was grouted when installed and the grout continually comes out and looks bad. Can I caulk over the old grout instead of regrouting or do I have to remove all of the old grout before doing anything.

"Can I caulk over the old grout instead of regrouting or do I have to remove all of the old grout before doing anything."

You need to remove at least enough of the old grout to get a decent bead of caulk into the joint.

Change in plane that involve different support structures should be caulked.
The supporting structures are very likely to move differently, and grout cannot absorb any relative movement without failing.