Burke as natural law theorist
A proper first step is to note with Russell Kirk that :
[Burke] takes
for granted a Christian cosmos, in which a just God has established
moral principles for man's salvation. God has given man law, and
with that law, rights; such, succinctly, is Burke's premise in all moral
and juridical questions. The religion of Edmund Burke is a very
interesting topic which cannot be examined in detail here; but it
needs to be mentioned before any consideration of Burke's political
fundamentals ... God gives us
our nature, said Burke, and with it he gives us natural law. But
that law, and the rights which derive from it, have been misunderstood by the modem mind - thus Burke continues:
The rights of men, that is to say, the natural rights of mankind,
are indeed sacred things; and if any public measure is proved mischievously to affect them, the objection ought to be fatal to that
measure, even if no charter at all could be set up against it. If these
natural rights are further affirmed and declared by express covenants, if
they are clearly defined and secured against chicane, against power, and
authority, by written instruments and positive engagements, they are
in a still better condition: they partake not only of the sanctity of the
object so secured, but of that solemn public faith itself, which secures
an object of such importance .... The things secured by these instruments and positive engagements may, without any deceitful ambiguity, be very fitly called the
chartered rights of men.("Speech on Fox's East-India Bill," Works of Burke (Bohn ed. II: 176.)
R. Kirk, 'Burke on Natural Rights', The Review of Politics, Vol. 13, No. 4 (Oct., 1951), pp. 441-456: 441-2.)
If Burke's politics is set within this natural law framework, it clearly has some defence against the charge of 'cultural relativism'. Natural law is not relative to culture.
Burke as pragmatic conservative
Burke can and does accommodate an element of historical and cultural specificity, however. He sees the common law rights and associated political arrangements and conventions of his fellow citizens as embodying, firmly but not perfectly, the moral principles mentioned above and as congruous, consonant, with the Christian cosmos. He does not hold that exactly and only the rights, arrangements, and conventions that he defends, circa mid-18th century Britain, are universally prescriptive for politics. Other societies may accommodate the relevant moral principles under different political configurations. He does not criticise pre-1789 France for failing to replicate a British style of politics. One size, to apply a modern phrase to an 18th-century situation, does not fit all.
A further sense in which Burke accommodates historical and cultural specificity is disclosed in his sense that social and cultural change is a fact. The proper response to this fact is not automatically to resist such change or to slow it down (to adopt what one might term a 'reactionary' atttiude) but to initiate and support as much change as is necessary to preserve what is essentially valuable in the rights, arrangements, and conventions that we have. This flexible and pragmatic reponse is clear in Burke's observation: 'A state without the means of some change is without the means of its conservation' (E. Burke, Revolutionary Writings: Reflections on the Revolution in France, ed. I. Hampsher-Monk, Cambridge: CUO, 2014: 23). The requisite change will inevitably be adjusted to circumstance; it cannot avoid being historically and culturally specific.