I do not believe that cowardice or bravery exist - in my opinion. They are both words that are used to pressure people into doing what you want them to do; such as when a kid at school pressures their peers into trying drugs, and if they refuse they will be called a "chicken". For the sake of argument, though, I will imagine that I do believe these 2 concepts exist for a moment:
Cowardice (by the definition) is fueled by fear; humans have a fear of death which is part of our survival mechanism. To commit suicide requires a person to pass through that fear, and not to avoid it. Even if human beings did not have a natural fear of death this point would still stand; since not everybody who has taken their own life does so because they were frightened to live - they were just tired, or in pain, or for any number of other reasons.
If anything it is "cowardly" to be against suicide. People oppose it because they are afraid to confront the harsh reality that life can be unimaginably horrible for certain individuals, so much to the point that no amount of generic platitudes will improve the sufferers state. Every time a person witnesses or hears about another human taking their lives it reminds them of just how fragile their own bodies are, and that at any point they could be rendered crippled beyond repair too; in turn this threatens to shatter the illusion that life is always worth living no matter what, and it makes them very, very uncomfortable.