Infinite simple Galois groups

Any profinite simple group is finite, since it has nontrivial finite quotients (the conjugates of a finite index subgroup intersect in a finite index subgroup).

Th notion of simple is not very interesting for profinite groups. The "right" concept is just infinite. We say that a profinite group is just infinite if all its non-trivial normal closed subgroups are of finite index. $SL_2(\mathbb{Z}_p)$ is an example of a just infinite profinite group. Another example coming from Galois theory is the Nottingham group which is an open subgroup of index p-1 in the automorphism group of the field $\mathbb{F}_p((t))$.

There are many examples of just infinite profinite groups. There isn't much general theory except Wilson's dichotomy that they are either Branch Groups (e.g. Grigorchuk group or the Gupta-Sidki group) or they contain an open subgroup which is direct sum of hereditarily just infinte profinite groups (i.e. every open subgroup is also just infinite) and some results of Colin Reid.

One way to intepret the question is: if G is a simple p-adic Lie group, like SL_2(Z_p), do we expect there to be an extension of Q with Galois group G? (Here the point is that G is not literally simple as a group, but it has no positive-dimenional p-adic analytic group quotient.)

GL_2 is easy -- adjoin the Tate module of an elliptic curve -- but is not simple. Maybe you can get SL_2(Z_p) via Shih's construction (as described e.g. in Serre's "Topics in Galois Theory"?)