Yes, as long as you are adding a verb after "carry on" as a direct object, you need to put it in the gerund "-ing" form.
This is because "carry on" requires a nominal (noun) object.
E.g.: He carried on *the conversation* ("the conversation" is a noun.)
She carried on *the family business*
Now, when you have a verb that follows "carry on" you nominalize it (make it into a noun) by using the gerund form of the verb, by adding -ing.
E.g., He carried on his singing.
The doctor carried on caring for his patients despite being sick himself.
Of course, if the verb is no longer a direct object of "carry on," then you do not necessarily have to use the gerund "-ing" form.
E.g., He carried on as if nothing happened.