Let $F: \mathbb{R}^n \to \mathbb{R}^n$ be a differentiable map.
-editted- Let $x^\star$ be a fixed point of $F$.
Then, is it true that the fixed point iteration $x_{n+1} = F(x_n)$ converges locally if and only if the spectral radius of the Jacobian $J_F(x)$ at $x = x^\star$ is less than 1?
No. If you have a fixed point and the Jacobian at this fixed point has a spectral radius smaller $1$, then the fixed point is stable or attracting.
There may be many other, non-fixed, points where the Jacobian has a small spectral radius. Take $f(x)=x/2$. Then $f'(x)=0.5$ everywhere, but only $x=0$ is a fixed point.
Answer to corrected question: If the spectral radius is smaller than 1, then as written above, the iteration converges for all initial points close to the fixed points.
The other direction is not as clear, even if one direction is repulsive, the iteration could still lead to an attracting direction so that the iteration ends at the fixed point.