(1) Determined to catch up, he quickly got up and ran toward the finish line.
(2) Determining to catch up, he quickly got up and ran toward the finish line.
Yes, they are both grammatically correct. They show no practical difference in meaning, though "determining" in (2) is rarely heard nowadays, with "resolving" perhaps being more likely.
They are different constructions, of course.
In (1) "determined" is an adjective and the underlined expression is thus an adjective phrase functioning as a predicative adjunct. We understand that 'he was determined to catch up'.
In (2), by contrast, "determining" is a gerund-participle verb and the underlined expression is thus a clause functioning as a 'depictive adjunct' giving descriptive information about the referent of "he". It is interpreted with progressive aspectuality (the semantic category of aspectuality, not the syntactic category since it is pot possible to insert the progressive auxiliary here).