Disclaimer: I don't own Doctor Who. The title is from the Simon and Garfunkel song "The Sound of Silence." Many thanks to my beta, ElementalLiz.
When the Doctor thinks up his plan to fool the Silence, he does consider having one of the others stay with Canton. He knows it will probably be less dangerous than investigating the Silence alone. Rory would certainly appreciate him giving that job to Amy. But he doesn't (can't) do that. He is who he is because of his memories, even the ones of minor events. Because the Silence can remove his memories, they can change who he is, and so they frighten him more than he will ever admit. By staying with Canton, he won't lose any more memories. He might be a prisoner, but he'll be safe in every way that matters.
They don't know it, but just admitting to his friends that he has no idea what the Silence looks like is humiliating. He is a Time Lord; no other species should have the power to affect his memory. Every time he looks at them, he remembers so much. He's seen them many, many times – and forgotten about them many, many times. He has forgotten the first sight he had of Susan after her first day of school in 1963 because of them. He has forgotten snippets of the short amount of time he spent with Rose and Donna because of them. It makes him so angry. By removing his memories, even ones that last only a minute, the Silence are changing who he is, and that's one thing he can't forgive.
He knows he should warn the Silence about his plan and give them a chance to leave Earth. The Silence have caused deaths, yes, but so have the other aliens who he tried to convince to leave Earth. He's given Daleks that chance, was willing to die to let the Sontarans have it. But he doesn't give it to the Silence because he's too afraid of what will happen if they agree. He tells himself that he can't take the chance that he'll forget about the Silence and they'll take over another planet. That's true, but he's also terrified (him!) of what memories he'll lose if he runs into them again (and with his luck, he knows he will).
So he tells the humans to kill them. Because it is him saying it, even if the captured Silent is the one who actually speaks the words. It had no idea that Canton was recording what it was saying, that its words would be used to annihilate its entire species. He could have thought of another plan, one that didn't involve mass murder. He hates guns – has refused to use them since the Time War. He once said to the man who murdered his daughter that he would never shoot anyone. Now, he's no better than the Silence. He was mad at them for letting humans die, but he's letting (forcing) humans kill them. It's his fault every time one of them dies.