That would be really unnecessary and I don't even think it is possible. React is not like jQuery or bootstrap. It is not meant to make your page look nicer or render plain html. It can render provided html but then you will lose all those cool react features. React doesn't use templates and it is itself a view engine. I have seen people use it in replacement of backbone view and use it backbone collections/models but I don't think it would be possible using with a template engine.
Also keep in mind that, jsx translates to js objects. so giving react a pure html would not return an object and will be totally outside of react scope.
the only way I can think of, which would be really bad practice, is creating a new react-dom render in all your pages, and create an internal json ( yes you have to provide data to react, can't just read from your database etc ) object in each instance to provide some props.