Wikipedia says, "Group objects in the category of algebraic spaces over a field are schemes." There is no citation, but in an arXiv paper, arXiv:0907.3880, the claim is cited to the original paper of Artin, "Algebraization of formal moduli".
Looking over the other answers provided, I gather that the question is analogous to asking, "Is a group object in the category of orbifolds a Lie group?" Yes, because it can't have a stratification. It is apparently similar but more abstract in the setting of stacks (say). Anton's other question about a coset space $G/H$ is presumably a similar phenomenon. But I would guess that a double coset space $H\backslash G/K$ can be a stack, since in the Lie group setting it can be an orbifold.