No, one cannot.
Symplectic integrators guarantee reversibility and volume preservation (at least provided that the gradients, however they may be defined, are deterministic along a numerical trajectory). Reversibility and volume preservation then enables state selection procedures that use only the value of the Hamiltonian (for example with a Metropolis acceptance procedure between the initial and final states in Hybrid Monte Carlo or the multinomial scheme of Stan’s dynamic Hamiltonian Monte Carlo sampler). But this just ensures that the target distribution is one stationary distribution of the resulting Markov transition distribution.
That is not enough to ensure that Markov chains converge to that stationary distribution asymptotically let alone preasymptoticaly. As I discussed above the preasymptotic performance of Hamiltonian Monte Carlo that is so celebrated depends on how well the gradients match with the Hamiltonian used in the state selection scheme.
Again messing around with the gradients deterministically does not obstruct the right stationary distribution, but it does obstruct the finite time performance of Hamiltonian Monte Carlo. In order to track the posterior the sampler needs to be able to account fo this discontinuities.
Approximate gradients based on numerical solvers can be used successful with Hamiltonian Monte Carlo when those approximations are sufficiently accurate, which cannot be taken or granted.