NP hardness through Restriction
No, it is not true in general.

A general problem being NP-Hard does not imply a specialization will be NP-Hard. It is true the other way round.

For instance, MAX-CUT is NP-Hard for general graphs, but for the class of planar graphs, it is in P.

In your case, the $d$-regular graphs are a special case of graphs of max degree $d$, so NP-Hardness for max degree $d$ is not an automatic proof of NP-Hardness of $d$ regular graphs.

In fact, for a concrete example, look at this cstheory question: Is there a problem that is easy for cubic graph but hard for graphs with maximum degree 3? with the following answer(by David Eppstein) in the affirmative (copied for convenience):


  Here's a reasonably natural one: on an input $(G,k)$, determine
  whether $G$ has a connected regular subgraph with at least $k$ edges.
  For 3-regular graphs this is trivial, but if max degree is 3 and the
  input is connected, not a tree, and not regular, then the largest such
  subgraph is the longest cycle, so the problem is NP-complete.