Democracy does not just involve the process through which we determine leaders and make collective decisions. Democracy means protecting certain inalienable rights. It means guaranteeing individual freedoms. In this regard, an unamendable constitution furthers democracy.
At its core, democracy is about preserving individual rights and freedoms. Unamendable constitutions provide a far better mechanism for preventing human rights abuses and restrictions of civil rights than amendable constitutions. Therefore, unamendable constitutions are not just compatible with democracy, they further democracy. Unamendable constitutional provisions are supposed to function as a barrier to change. This barrier of change is needed because there are aspects of the constitution that should be protected from alteration. Amendments that are made in the constitution are because of the majority. There are aspects of the constitution that are supposed to be protected from the whims of the majority because it reflects the nation's identity. 
There are some aspects of the constitution that should be ammended. Certain states may be too reluctant to seek out more democratic methods of effective constitutional change. Parts of the constitution, specifically Article V of the constitution, can be subject to rigidity as well as being inconsistent with democratic legitimacy. Many have been critical because Article V makes the constitution difficult for the majority to ammend the consitution, something which is considered to be a tenement in democracy. 
[P1] Democracy is not reducible to legislative processes. It also encompasses the preservation of human and civil rights and the guarantee of individual freedoms. [P2] Unamendable constitutions are better than amendable constitutions at protecting human rights and individual freedom. [P3] Therefore, unamendable constitutions are more democratic than amendable constitutions.