The Spanish structure [disfrutar + verb-infinitive] should be translated as [enjoy + verb-ing] in English, and vice versa, in general.
There are some English verbs, like like, prefer, etc. that can take both to-verb or verb-ing. In general these are both translated using the Spanish infinitive. If there's any difference in meaning between the two English forms, it'll be lost.
I understand that your doubt stems from the fact that in English the -ing form (or gerund) can often be used as a verbal noun, while in other cases it functions as the present participle. The Spanish gerund (-ando, -iendo) doesn't do such double duty. Sometimes (as explained in another answer) you can have a gerund after some verbs like disfrutar, but it's not that usual, and the meaning is a bit different (in this case the gerund acts a complement of manner).
You cannot use a gerund after most of those Spanish verbs that can take an infinitive. In the following examples, the first sentence is OK and the second is wrong:
- Me gustó hablar contigo.✓ | Me gustó *hablando contigo.✗
- Prefiero comer solo.✓ | Prefiero *comiendo solo.✗
- Elegimos no tener hijos.✓ | Elegimos no *teniendo hijos.✗
Unrelated to the actual details of your question, but having to do with it in a sense: yes, you can use the gerund (-ando, -iendo forms) with at least one verb other than estar using the same structure:
- Ya vamos llegando. = "We're getting there already."
- Se fue derrumbando de a pedacitos. = "It fell apart by bits and pieces."
Note again that [ir + gerund] has nothing to do with [ir + infinitive]! The former is used for an ongoing process, while the latter is a future construction.