Cough and catching cold are symptoms of inflection of the upper respiratory tract. Red eyes are caused by conjunctival congestion. Because the upper respiratory tract inflection can lead to conjunctival congestion, people, therefore, may cough with the symptom of red eyes. Red eye does not mean pink eye. Pink eye is a kind of contagious eye disease. can be caused by exposing to bacteria, virus, allergy and mold and the spreading of inflammation of the nearby tissue. As far as I'm concerned, your illness should not be pink eye. You can take some cold medicine. That will be enough.
Yeah, it is possible that coughing may cause red eye or pink eye, especially pink eye. Redness in eyes does not mean you may or already get pink eye, or conjunctivitis. We have known that coughing is commonly due to cold. While cold is most possible because of bacteria and virtues. And it is known that pink eye or conjunctivitis is most possible to bacteria and virtues enter eyes. That is, bacteria and virtues can cause eye infections, pink eye is one of them. By the way, pink eye is not serious and you will be fine without any treatment options in 4-7 days. And don't rub your eyes with your dirty hands, even you think you have already wash them up.