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It's Renly's right to name his successor, provided said successor either marries into house Baratheon or bears Baratheon blood. Other Renly hasn't married into house Baratheon and definitely doesn't have Baratheon blood.There's also the possibility that Stannis get convinced, Renly abdicated willingly in favor of New Renly, he will grit his teeth over that a whole lot, but it's Renly's right to name his successor, of course this is only if we manage to convince Stannis that's what happened.
@tarrangar, by that logic you're saying that Stannis would keep his mouth shut if Renly decided to name Loras Tyrell his successor because it's Renly's "right" to choose his own heir.