The definition of a monoidal model category requires that the tensor product is biclosed (such is needed to ensure that the tensor product is derivable in both variables). Obviously, in the situation of relative categories and simplicial categories, which we know model all $\infty$-categories, we can't really pull this off.
So a few questions:
What explicit data, if any, give $E_1$-monoidal structures on simplicial and relative categories?
Given an actual monoidal model category $M$, an on-the-nose braiding satisfying the strict coherences gives $M$ an $E_2$ monoidal structure. Are all $E_2$ enhancements represented by strict braidings?
More generally, can we say that a monoidal model category $M$ is $E_n$ monoidal by demanding that particular explicit function complexes are $n$-connected?