Can you pigtail two sets of wires leading to two separate outlets and attach the pigtails to the GFCI's load terminals?

 

Yes you can. Pigtailing would be the recommended method.

There is also a place to connect two sets of line wires or two sets of load wires directly to the device but it makes installing the device difficult.

 

Thanks very much for that.

It turns out that the box I want to add a wire to is fed from an ungrounded receptacle. Geez! So I installed the GFCI there reasoning it will protect the downstream outlets. Two things though. I wire-nutted the load wires in the ungrounded box since I may have an issue with the box it fed. The GFCI tests correctly reset button-wise but all three lights on my 3-pronged tester illuminate (one not as brightly as the other two) when I plug it in. Then this morning a lamp on the circuit dimmed momentarily. So I uninstalled the GFCI and wire-nutted the line wires too, thinking I should have a pro look at it. Does it sound to you like I have an issue with that circuit or the GFCI? The tester lights up as expected on my other GFCI and non-GFCI outlets.

The other thing is...when I looked at the box I want to run another outlet from the two black wires were pigtailed and the pigtail was attached to both gold screws on the receptacle. The white wires were attached normally. Why would that be? (That outlet tested as ungrounded too, although there was a ground wire attached to the receptacle when I tested it.)

I'm thinking this install is above my pay grade, but it would be nice to understand what is going on.

Thank you again.