So... One of the postulates of special relativity is that the speed of light is constant in all inertial reference frames. (General relativity does not overrule this, it just lets us consider non-inertial reference frames, and whilst gravitational waves do not appear in special relativity, since SR still applies, we can still consider them to be bound by its effects).
From SR we can work out a tonne of stuff, such as the fact that, no matter how fast we travel, we can never catch up with a photon, and we can work out that it takes an infinite amount of energy to accelerate something of finite rest mass to the speed of light. In even more generality, nothing carrying information can travel faster than the speed of light (since you can show that information itself implicitly contains some concept of energy and hence mass).
From this alone, we should be able to see that there's nothing specific to a particle rather than a wave allowing it to travel at the speed of light.
Speed in waves typically arises as a result of the medium in which they travel, with interactions in the medium slowing them down. Since optical and gravitational oscillations can both travel in a vacuum, without being held back by a medium, it seems reasonable to assume that they should both travel as fast as possible (or at the very least at the same speed). We could further this argument by discussing the concept of rest-mass for a wave travelling in a vacuum, which would be likely to make this clearer.
Basically, I don't know enough GR to prove this rigorously to you, but from your question, I'm guessing you don't have a huge amount of experience with SR (I genuinely do not mean this as a put-down, it's a good question!), and so I'm guessing that even if I could it might not make much sense. Without that, I'm going to have to trust the fact that gravitational waves were predicted pre-GR, and my general intuition on waves. And of course, we can't prove this either way, because this is physics, but suffice it to say that physics, as it exists today, predicts gravitational waves that travel at the speed of light, without saying anything about whether or not they are quantised to particles.