Is eating fish eggs the same as killing? Is eating fish eggs that we normally find in sushi dishes the same as kiling? Is there any life terminated when we eat them?
the biggest problem is that the fish species for which there is a caviar market are becoming extinct because they cannot reproduce. Many countries are trying to ban the import of the eggs to protect the species.

there are many fish for which the eggs are sold, for this question there are two categories: 

1) the eggs are harvested from inside the mother fish. She is pulled out of the water, then to retrieve the eggs her body is sliced from tail to mouth to access the eggs, usually on the boat, often while she is still alive. The eggs may have been fertilized inside her (depending on species).  

2) the fish lays the eggs, which are fertilized (either inside her or in the water), and then they are harvested. 

either way there is death.