Finish cooking a pork roast next day after partially cooking it

Every Kitchen needs a meat thermometer. Little too late in this case but it would have saved you here. I never used one at home until after I started working in a restaurant that only had a flat top and a convection oven with burgers on the menu. I got a relatively cheap one for $5-10 at Wal-mart and it does the job. Another option for next time is to check your meat often and poke it with a spoon, once it feels firm throughout, like the skin on the back of your hand when you make a fist, it's done.

It's hard to say from the picture, but pork fat when cooled in a refrigerator does not have the texture of rubber. It is firm but stays deformed when you squish it. What does have the texture of rubber and could have come from the pork is gelatin. When you cool certain cuts of beef, pork, or other meats you break down the connective tissue and dissolve the gelatin in the cooking liquid or meat juices. If you cooked it, cut it, and then chilled the cut pieces they likely let out some of their juices and those juices then set (gelled) in the refrigerator due to the gelatin.

I would just heat it up, and the gelatinized juices will become liquid again.