What is the meaning of "absolutely-ed" in this sentence?
The author is making use of the ability to make any part of speech into a verb. This is known as verbing (or verbification), and usually takes place with nouns. In this case, absolutely is used as an interjection; when an interjection is verbed, it usually refers to saying that interjection. Other examples of verbing:
Therefore, absolutely-ed (or perhaps more properly, “absolutely”-ed) is the past tense of the verbed quotation “absolutely”, which means to say “absolutely”, and is derived from the interjection/adverb absolutely.
It's important to note that it's still very uncommon to verb words like absolutely or yes or hey. When quotations such as this are turned into verbs, it's most often in commentary on something a third party has just said:
On a more common note, there are countless verbs in English that come from nouns.
It means he said the word "Absolutely!" and that he did that with real gusto.
("Heartily" means with feeling, loudly, from the heart.)