Bloodshot eyes are eyes with noticeable blood pigments and swollen blood vessels. You generally get bloodshot eyes from lack of sleep.

no, it would not cause your eyes to be bloodshot. Bloodshot eyes come from inflammation of the conjunctiva (the mucus membranes over your eyes/eyelids). Thinning of the blood could not cause conjunctivitis (bloodshot eyes) unless you have an allergy to that medication that could cause bloodshot eyes (which I would think would be almost impossible).

The cast of With Bloodshot Eyes - 2013 includes: Andrew Nance as Man on Phone

very, very, painful bloodshot eyes along with indigestion, headache and a stuffed up nose.

Yeepp