How should I seal a 2 inch gap between my driveway and garage slab?
I'm not sure caulk is the right approach in the first place for a gap of two inches. There are commercial products that do a good job (seen in cases such as pre-cast building walls), but you'd need the appropriate applicator tool and large quantity of material. Typical caulks are only suitable for gaps of up to half an inch or an inch, depending on the product. I'd be surprised if any claim usefulness to two inches.

I think I'd use asphalt patch mix. Clean out the channel to 3" of depth, rinse and compact with water, and install using a suitable tamping tool (a sledge hammer and 2x4 on end to set, flat to finish). 

If both of your slabs are concrete, consider renting a diamond saw, cutting the channel to 12" or so, and pouring a proper slab segment with a few bags of ready-mix.