Walking to infinity on the primes: The prime-spiral moat problem

I don't know if any of the probabilistic or percolation models related to the Gaussian (Eisenstein etc.) prime walks have analogues for the problem you suggest. However note that if such an infinite walk was possible it would imply that the gap between successive primes would be $O(\sqrt{p})$, i.e. it is a (slightly weaker) form of Legendre's conjecture. Also note that this is stronger than the bound on prime gaps implied by the Riemann hypothesis which is $O(\sqrt{p}\log p)$, so no, the problem you suggest is not any easier than the other conjectures about patterns of primes.

Percolation theory suggests that the probability one can one can walk to infinity is 1 if the density of randomly chosen stepping stones is at least a certain critical number, and is 0 if the density is less than this number. Since the density of primes in the Ulam spiral and the density of Gaussian primes in the plane both tend to zero, the density of stepping stones is 0. This suggests that one cannot walk to infinity on either primes in the Ulam sprial or Gaussian primes, for any bounded size of step.