  Let $M$ be an open Riemann surface. It was proved by Alarc\'on and
Forstneri\v{c} (arXiv:1408.5315) that every conformal minimal immersion
$M\to\mathbb R^3$ is isotopic to the real part of a holomorphic null curve
$M\to\mathbb C^3$. In this paper, we prove the following much stronger result
in this direction: for any $n\geq 3$, the inclusion $\iota$ of the space of
real parts of nonflat null holomorphic immersions $M\to\mathbb C^n$ into the
space of nonflat conformal minimal immersions $M\to \mathbb R^n$ satisfies the
parametric h-principle with approximation; in particular, it is a weak homotopy
equivalence. We prove analogous results for several other related maps, and we
describe the homotopy type of the space of all holomorphic immersions
$M\to\mathbb C^n$. For an open Riemann surface $M$ of finite topological type,
we obtain optimal results by showing that $\iota$ and several related maps are
inclusions of strong deformation retracts; in particular, they are homotopy
equivalences.
