Ken's answer--that the adverb fully modifies the verb is--is almost, though not entirely, correct. Fully modifies not just the verb is, nor does it modify just the possessive pronoun his. It modifies the entire verb phrase "is his." For a fuller explanation, please keep reading.
We need to remember that adverbs are a notoriously difficult category to capture in a definition. Quirk et al.* provide a good description of adverbs:
"Because of its great heterogeneity, the adverb class is the most nebulous and puzzling of the traditional word classes. Indeed, it is tempting to say simply that the adverb is an item that does not fit the definitions for other word classes." (p. 438). Therefore, the behavior of certain adverbs is very hard to pin down.
Surprisingly (and shockingly), none of the leading grammar references has any mention of the combination
Grammatical subject + be + degree adverb + possessive pronoun
...as in Ken's example. All the references treat degree adverbs in great detail when they occur as modifiers of verbs, as in
I fully agree with him
She completely forgot their date
or as premodifiers of adjectives and adverbs:
...his thoroughly disgusting table manners
You gambled away the money terribly fast!
However, these major references have nothing to say about their occurrence in the verb phrase with the verb be plus a possessive
pronoun as subject complement. (The verb phrase is the verb plus everything with it that completes its meaning, including subject complements, as in Ken's example.)
Swan states "Adverbs of degree (like nearly, almost, etc.) cannot go in end position." I think he didn't mean all degree adverbs (although his punctuation suggests that idea).
It is true that a large number of degree adverbs, like almost, downright, and nearly, as well as relatively, rather, and quite (plus many others) do not occur in final position. However, a great many "highest-degree" adverbs, e.g. fully, altogether, absolutely, completely, thoroughly, utterly, and wholly, DO occur in final position.
One of the major references, Huddleston and Pullum et al.,** explains that high-degree adverbs may indeed occur at the end of a clause as well as in the middle (italics mine):
"The [highest-degree adverbs] can generally be positioned centrally or at the end of the clause, reinforcing their height on the scale of degree.... In end position they will typically carry the stress..."
(p. 722). Examples they provide are
I agree with you absolutely
This ruined the evening totally
("Final position," by the way, is just that: the last item in the clause. In Ken's sentence "It was almost a home run," the item in final position is the noun phrase "a home run.")
The adverb fully and other similar high-degree adverbs modify the entire verb phrase. We can see this if we move the adverb fully to the end and insert a comma:
The responsibility is his, fully
Other similar highest-degree adverbs don't even need the comma (I don't know why fully does need one)
The responsibility is his entirely/totally/completely
Further examples are
The house has become totally ours/ours totally!
The fault is not entirely mine/mine entirely
It seems clear, then, that adverbs of high degree in this kind of
construction modify not just the verb, but the entire verb
*A Comprehensive Grammar of the English Language (Longman, 1985)
** The Cambridge Grammar of the English Language (Cambridge University Press, 2002)