This is a question Gail asked in a comment. She is a well known figure in the cat world ;). She couldn’t find a clear answer on the internet. I searched myself and had similar problems. This is an attempt to clarify the situation. Right away I can tell you that despite extensive searching I have failed to find a definitive answer from a sound source. Nonetheless here is my answer.
Whether conjunctivitis is contagious from cat to dog depends on the cause. There are various causes of conjunctivitis. This is one reason for confusion. The two main causes of conjunctivitis are viruses and bacteria. These are two different sorts of organism.
Conjunctivitis is sometimes referred to as “pink eye“. It describes inflammation of the membrane covering the back of the eyelids and the surface of the eyeball. When inflamed it looks pink.
A veterinarian website, vetinfo.com says in relation to conjunctivitis:
“The transmission from cats to humans can also be made by direct contact…”
That implies the same could occur between cat and dog because it implies that the bacteria can be transmitted between different species. One of the causes of conjunctivitis is a secondary bacterial infection on the back of the primary infection, which most commonly is the feline herpes virus. There might be pus and mucus (thick yellow to green discharge) forming a crust. This is purulent conjunctivitis. This suggests a secondary bacterial infection as the cause.
Bacterial infections are contagious, but only when the animal with the bacterial infection is in direct contact with the animal which is healthy. That is my understanding of the way bacteria is transmitted.
So a dog would have to make direct contact with a cat with conjunctivitis. Could a dog licking an infected cat over the eyes get the disease? It seems the answer is yes, it is possible.
The feline herpes virus is not zoonotic. “Zoonotic” means a disease that can be transmitted between different species of animal including animal to human and vice-versa. So if the cause of the conjunctivitis is entirely due to the feline herpes virus, it could not be transmitted to a dog.
Accordingly, if a cat has a feline herpes virus infection and conjunctivitis as a result (as a secondary bacterial infection) this condition could be contagious to a dog, if the dog is in direct contact with the cat in some way or other.
The direct contact may be indirect in fact. What if a person touched the cat’s eyes, trying to clean them and then touched the dog’s eyes? That would seem to be possible and it could transfer the bacteria.
Another form of conjunctivitis is “serous conjunctivitis”. This is caused by irritants such as pollen and wind. This condition is obviously non-contagious.
This is a difficult subject. Answers are hard to come by, or unavailable even in good books. I think it is correct to say that transmission of viruses between different animal species is rare, while transmission of bacterial infections between different species is less rare. But viruses can be transmitted through the air and in other ways, while for bacteria there has to be direct contact. Another example of bacterial transmission between species (cat to human) is toxoplasmosis.
OK, the answer to the question in the title is yes, possibly if the conjunctivitis is caused by bacteria.
Associated: Feline conjunctivitis