Actually, he couldn't. He admits in his documentary that he and his father did not have a great relationship and he didn't know or get what Star Trek was about (hence the narrative of his documentary which was quite painful to watch at times). He was like 16 or 17 when Gene died therefore he could only speak for himself and not his father.
As for how much legal authority Rod Roddenberry has to creatively direct Star Trek? I would guess none. CBS has a duty to pay him his father's percentages (like 5 or 10% but part of that was watered down by divorce), however, they can produce Star Trek without his input.