Can I hook up under-sink water filter to sprayer outlet on faucet?
The diverter valve built into your faucet is downstream from the faucet valve that is designed to hold back "line" pressure. The diverter shunts water toward the spray head when you open the spray head valve, and directs it toward the fixed spout otherwise. 

The faucet sprayer head/nozzle is not designed to hold back system pressure. They would have to be more robust to the point of being unwieldy to do that. Also, the main valve is where hot and cold are mixed, giving you temperature adjusted water at the sprayer head. If you hook directly to the spray head it will leak.