What is the ruling on a fasting sick person who takes a blood-thinning medication every 12 hours, and is it permissible for him to fast for only 12 hours, or should he break his fast?

Fasting for half a day is not valid. Fasting is valid with the intention from dawn until sunset. If a patient cannot find an alternative medication that is taken once instead of the one used twice, then there is no sin upon him if he breaks his fast, and he must make up for it later.