Yes, both the present and past participles are grammatically valid here, but neither would be as common as the infinitive.
The progressive aspect usually indicates an ongoing action. However, it seems unusual for a person to be falling over an extended duration of time. Furthermore, the sentence implies that the first person is falling, too, but if they are falling at the same time, then why would this person be labelled the "second" (which implies that his falling takes place afterward)?
The past participle would not be as unusual but would still be less likely than "He is the second person who has fallen this morning." The past participle would be more likely for a transitive verb, e.g.:
He is the second person saved by Superman this morning.
He is the second person felled by Lex Luthor this morning.
(Thanks to Paul Tennenbaum for suggesting transitive "fell" in a comment.)