For a start, the present continuous is not used in your example, which bring us to the explanation of a verb not having a tense.
That -ing verb gives the contrasting effect; in this case: the consequence of the previous action - so, it bears adverbial meaning : consequently, therefore, thus; and could be called "a complex sentence". Your sentence can be rephrased, but it will not have or establish the same syntactic construction.
Some adverbial clauses can be put into front position of a sentence. But they will not necessarily carry the same or, even, similar meaning.
Faced with a serious problem, I decided to walk away. (Here, the past
participle of the verb: "faced" has the passive sense that also
implies adverbial meaning - Because I was faced with ....... );
Now, if you say the following sentence:
I decided to walk away, faced with a serious problem.
This sentence absolutely means the same as the latter sentence. But this is often not possible with dependent -ing clauses.
As in your example:
The man in the wheelchair pushed himself away, [and therefore (he left) ] leaving the card on the table.
If the comma is removed, the meaning will not make much sense. In fact, it would indicate "how the man pushed himself away" and obviously modify the verb, which is not intended here, and reversing the position would specify the time adverbials : when, while.
It is the comma that makes the difference in the sentence. Also, this is a common contrasting construction. Constructions that can imply reasons, purposes, conditions, some time adverbials and so forth can easily be understood or determined for the independent clause they follow.