It is a result of May's work on operads that the homotopy category (or $\infty$-category, if you prefer) of connective spectra is equivalent to a full subcategory of the category of representations of the little $\infty$-disk operad on the category $Top_+$ of pointed topological spaces (the full subcategory is then cut out by the condition of having inverse up to homotopy). Here the symmetric monoidal structure on $Top_+$ is given by smash product. This can be thought of as the homotopy-theoretic analogue of a definition of abelian groups as sets with some additive structure.

It is also of course true that the category of associative (or $E_n$) ring spectra is defined as representations of a suitable operad on the category of spectra, with symmetric monoidal structure the tensor product $\wedge$ operation. If we require the underlying spectrum to be connective, then (as far as I understand), we can work with this definition without leaving the connective-spectra category.

My question is whether there is some sort of generalization of an operad structure that gives ring spectra directly from $Top_+$, in a way similar to the definition of rings as sets with two operations.

The difficulty with this seems to be that the $\wedge$ operation is defined using some universal property (and, as far as I understand, doesn't admit a nice direct description on the level of $\Omega$ spectra): I'm curious if there are ways to get around this.