Rust programs have bugs. Rust programs have security bugs. Are they mediated by memory usage bugs? Probably not, unless the program has unsafe blocks, or uses libraries with unsafe blocks, or libraries that use libraries that have unsafe blocks, or call out to C libraries. Or tickle a compiler bug.
Can it leak my credentials to a network socket as a consequence of any of those bugs, memory or otherwise?
Putting your memory errors in unsafe blocks may make them invisible to you, but that does not make them go away.
So, yes, of course it can.