There's an interesting piece at ConservativeHome: Andrew Lilico: Should we care about the demise of the constitutional monarchy? which does provoke my above question.
The simplest answer would be "no but it looks otherwise". Both individual conservatives and conservative parties in many parts of the world are not seeking the establishment or restoration of monarchies and in some countries conservatives are actively seeking the end to monarchy (e.g. Australia).
In the UK some try to claim the Conservative Party is an inherently monarchist party, but given that the monarchy is never an issue in party politics here, no-one seems to have examined whether this support is just a traditional "it ain't broke, don't fix it", "it's not a priority to get rid of" or a genuine reverence for monarchy. When at meetings with other party members we've been asked to try to define what a "conservative" is, I've only ever heard one person out of many mention the monarchy at all in their response.
There is a fundamental difference between adherence to a concept for its own sake and resisting unnecessary change. Conservatism is traditionally the latter, not the former.