What single word (verb) means 'to assert as a moral norm' or 'to support as a moral norm'?

See espouse at MERRIAM WEBSTER, defined as

to express support for (a cause, belief, etc.)

to take up and support as a cause.

Sanctify

Is a good choice.

To make something socially or officially acceptable. (Cambridge)

The word also has religious connotations, but can be used in the secular sense, as in the above definition. The religious overtones work well with your requirement that the word should:

"... assert as a moral norm"

I would say.

Claim has definitions that carry the idea that there is a "right" or "entitlement" and that you are also invoking such right or entitlement as well as referencing it.

Websters Definition of claim transitive verb

1

a : to ask for especially as a right

claimed the inheritance

The driver claimed the right to a hearing.

b : to call for : require

this matter claims our attention

There are several words that come to mind, one of which has already been noted by @vickyace.

Here are some others that also suit your intended meaning:

maintain: to affirm in or as if in argument

...they often fall back to defend his earlier positions, in which he maintains the absolute rejection of all truth claims.

endorse: to approve openly, especially to express support or approval of publicly and definitely

...they often fall back to defend his earlier positions, in which he endorses the absolute rejection of all truth claims.

Alternatively, you could rephrase your sentence as follows:

...they often fall back to defend his earlier positions, in which he calls for the absolute rejection of all truth claims.