An adverb or adverb phrase, though it may modify an entire clause just as an absolute phrase does, fails to fulfill a basic criterion for that construction: it is grammatically impossible for an adverb phrase to have a subject.
The analogous construction, however, is the sentence adverb, a disjunct which takes the reader outside the argument of the sentence and expresses a resultant attitude or opinion. As the name implies, a sentence adverb does not attach grammatically to a verb, but, as an absolute phrase, to an entire clause.
Sentence adverbs, of course, need not be phrases, but for the sake of illustration, here are a few examples:
Unfortunately for researchers, people under the age of 65 have no identifiers other than their Social Security numbers and have no population-wide insurance system that could (like Medicare) be used to track medical events over time. — Tools For Evaluating Health Technologies, 2004.
A common construction in which an adverb phrase modifies a sentence or clause is adverb + enough:
Ironically enough, British victory in the Seven Years' War set the stage for the revolt, for it freed the colonists from the need for British protection against a French threat on their frontiers and gave free play to the forces working for separation. — Robert W. Coakley, Stetson Conn, The War of the American Revolution, 2002, 24.
What is a Society? Curiously enough, it is not easy to find social scientists who seem to know – and are ready to explain – what a "society” is. — Claus Offe, “Is there, or can there be, a ‘European Society’?” in: Ines Katenhusen, Wolfram Lamping, eds., Demokratien in Europa, 2013.
The kitchen, like the rest of the house, was small and very crammed. The largest room in the whole house was the bathroom, surprisingly enough. — Meagan M. Donohue, Sincarian, 2009, 139.
Since gracefully describes the manner in which an action is performed rather than express an authorial comment, it is difficult to imagine its use as a sentence adverb even in the same construction:
Nandini had made up her mind to go with them, had even changed into outdoor clothes, but she had backed out, gracefully enough, without rancour. — Ambika Sirkar, No Crystal Stair, 2011, 56.
Despite the parentheses and the adverb + enough phrase, gracefully enough still expresses the manner in which Nandini backed out, i.e., it still modifies the verb, not the clause, as, say, surprisingly enough, which would be a comment on her behavior and personality, not how she backed out of a planned engagement.