  It is generally believed that a dilute spin-1/2 Fermi gas with repulsive
interactions can undergo a ferromagnetic phase transition to a spin-polarized
state at a critical gas parameter $(k_{\rm F}a)_c$. Previous theoretical
predictions of the ferromagnetic phase transition have been based on the
perturbation theory, which treats the gas parameter as a small number. On the
other hand, Belitz, Kirkpatrick, and Vojta (BKV) have argued that the phase
transition in clean itinerant ferromagnets is generically of first order at low
temperatures, due to the correlation effects that lead to a nonanalytic term in
the free energy. The second-order perturbation theory predicts a first-order
phase transition at $(k_{\rm F}a)_c=1.054$, consistent with the BKV argument.
However, since the critical gas parameter is expected to be of order O(1),
perturbative predictions may be unreliable. In this paper we study the
nonperturbative effects on the ferromagnetic phase transition by summing the
particle-particle ladder diagrams to all orders in the gas parameter. We
consider a universal repulsive Fermi gas where the effective range effects can
be neglected, which can be realized in a two-component Fermi gas of $^6$Li
atoms by using a nonadiabatic field switch to the upper branch of a Feshbach
resonance with a positive s-wave scattering length. Our theory predicts a
second-order phase transition, which indicates that ferromagnetic transition in
dilute Fermi gases is possibly a counterexample to the BKV argument. The
predicted critical gas parameter $(k_{\rm F}a)_c=0.858$ is in good agreement
with the recent quantum Monte Carlo result $(k_{\rm F}a)_c=0.86$ for a nearly
zero-range potential [S. Pilati, \emph{et al}., Phys. Rev. Lett. {\bf 105},
030405 (2010)]. We also compare the spin susceptibility with the quantum Monte
Carlo result and find good agreement.
