How do you mend a broken heart, or a broken tomato for that matter? You cant sew the tomato back together and it's not going to fix itself, but that doesn't necessarily mean you have to discard it.
Minor wounds will sometimes be grown over as the fruit matures, the rest of the tomato will sometimes envelope it with little or no harm discernible. Deeper wounds however are commonly attacked by insects as well as being highly susceptible to fungus and assorted bacterial pathogens.
If there is any sign of infestation on the Tomato get rid of it. If it looks like a fungus issue do not put it in the compost as the fungus may find the rest of your compost hospitable and re occur next season. If the damage is minor eating split tomatoes is okay. It's a good idea to pick these tomatoes as soon you notice the damage, ripe or not. Cut out the damaged section and put it to good use. I wouldn't advise using split tomatoes for any canning endeavors.
A tomato on the vine that is starting to show signs of getting ready to split should be harvested. Allowing it to ripen on the windowsill is a good idea. Leaving it on the vine will only make the tomato worse for wear and hasten the damage.