No, because the subject "Jonathan" is the doer of the action, not the recipient of it. "On the ice" is treated as a locative (or manner) adjunct. It tells us where/how he skidded.
(2) "Jonathan hurt himself skidding on the ice".
To express a meaning where the subject causes and receives the action, a reflexive pronoun is required. In this example, "Jonathan" is the antecedent for the reflexive pronoun "himself", so the meaning is that the person hurt was the same as the one who caused the hurt: "Jonathan hurt Jonathan".