Although modern grammar doesn't recognize the difference between gerunds and present participles, if we were to distinguish them, then in your case we would have a 'gerund'. The clearest way to see this is to note that you can replace the accusative him by the genitive his, and the genitive can only work with the 'gerund'.
On the one hand, as tchrist and Edwin Ashworth have said in the comments, modern grammar does not distinguish between gerunds and present participles. For example, CGEL has whole sections whose entire purpose is the defense of the proposition that those two are not distinct (pp. 82-83 and 1220-1222).
On the other hand, CGEL also says this (1220-1221; the asterisk, '*', means that what follows is not acceptable English):
A difference in internal form: case of the subject NP (noun phrase)
There is one respect in which 'gerund' and 'present participle' clauses differ in their internal form: with 'gerunds' the subject may take genitive case, with plain or accusative case a less formal alternant, but with 'present participles' the genitive is impossible and pronouns with a nominative-accusative contrast appear in nominative case, with accusative an alternant restricted to informal style. Compare, then:
 i She resented his/him/*he being invited to open the debate.
ii We appointed Max, he/him/*his being much the best qualified of the candidates.
This difference, however, is obviously relevant only to those constructions where the non-finite clause can contain a subject: it cannot be used to justify a distinction between 'gerund' and 'present participle' in the numerous constructions where no subject is permitted. In terms of our analysis, the contrast in the case of the subject is handled by our distinction between complement and non-complement gerund-participials: genitive case is restricted to the former, nominative to the latter. If the traditional distinction of gerund' vs 'present participle' is to be maintained, it must be based primarily on properties of the subjectless construction. But here there is no difference at all in the internal form of the constructions.
So: if the subject can be in the genitive, then what we have is a gerund. Now consider your sentence:
 We’re depending on his/him/*he finishing the job by Friday.
This is just like CGEL's [39i], and thus finishing is a 'gerund'.