I have an older 240V 1000W electric wall heater (no fan, built-in thermostat) which has 2 gray wires for attaching incoming supply. One gray line goes through the thermostat then on to one end of the heating coil itself. The other gray line goes directly to the other end of the heating coil. The existing supply is two 120V lines (each with a black, white, and ground). What's the proper way to connect the 2 blacks and 2 whites to the 2 gray lines of the heater?

The proper method will be to take one of the 120 volt feeds and change the 120 volt single pole breaker to a two pole 240 breaker, reconnect the black and white to the 2 pole CB, tape the white at both ends to re-identify the white now as a hot wire, and then connect the black and white to the two gray input wires at the heater.

This will leave the other 120 volt feed for some other purpose.

Also, You Cannot parallel conductors to increase Ampacity being smaller than 1/0.