No, unless... the predators limit their own population in order to match the natural death rate of the prey. As soon as there are more predators than there are naturally dead prey, that means either a predator is going hungry or a prey is going to be hunted.
How likely is it that the predator is going to go hungry? How likely is it that the predator society will "rebalance" their numbers by eliminating one of their own? Some would say not bloody likely at all.
Let's factor in environment and go to the logical extreme. Let's say the ecosystem is at max capacity for prey, i.e., the prey are at max possible population for that ecosystem. The predator population can grow to consumerconsume the number of naturally dying prey. Then add one more predator. Now the predator population is feeling some stress. How will it react to that stress?
There are many ways and many reasons that species limit their own numbers. A new alpha lion notoriously will kill off the cubs of the preceding alpha. Many species, say crocodiles, will indiscriminately eat the young of its own species. And of course there's always straight up cannibalism and of course, war. Chimps go to war. Ants go to war--even between colonies of the same species.
While none of these actions are specifically undertaken with the goal of controlling the population of one's own species, maybe this predator prefers to do one of these things before choosing to kill prey. (Which also would need some 'splainin.)