[susan53]

Grammatically, and in terms of meaning, there's no difference to the use of the gerund/infinitive after "continue". I'm going to change your example to the past, because it's easier to think iof a context.

I told him his dinner was ready but he didn't come - he just continued talking to his friends on facebook. So I gave it to the cat.

I told him his dinner was ready but he didn't come - he just continued to talk to his friends on facebook. So I gave it to the cat.

There, for me, the gerund sounds slightly more likely. That could, however, be a personal preference - when the language gives different ways of saying the same thing individuals often differ in the options they tend to choose most frequently - this can be due to regional background, gender, or a host of other sociolinguistic factors.

Sometimes too, the language itself pushes us towards one option or the other. In the examples above, the verb was in the simple form, but if it's in the continuous, so that there's already an -ing form, then the infinitive option is chosen to avoid having two -ing forms together - ie, to avoid ..

* Despite the crisis people are continuing investing in the stock market

...the infinitive would be used :

Despite the crisis people are continuing to invest in the stock market

But to get back to your original example - I don't think your feeling that "carry on" is more natural is actually connected to the grammar at all - it's a lexical and stylistic issue. Often, when there's a choice between a one-word verb (especially as with "continue" when it's derived from Latin) and a phrasal verb, neutral-informal English will prefer the phrasal verb. So if you compare ...

I told him his dinner was ready but he didn't come - he just continued talking/to talk to his friends on facebook. So I gave it to the cat.

I told him his dinner was ready but he didn't come - he just carried on talking to his friends on facebook. So I gave it to the cat.

... then I'd agree that "carry on" sounds slightly more natural. But it's the lexical difference that's important, not the grammar.