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It came to my attention in this question that I had an incorrect notion that the second verb could always be omitted when the sentence had a modal verb. For example:
Ich will nach Hause.
Ich muss dringend zum Arzt.
Ich mag nicht in die Schule.
Was soll ich damit?
Ich darf das nicht.
Ich kann Englisch.
But a counterexample where the “rule” breaks:
Ich kann Auto [fahren].
This doesn’t work when the second verb is omitted. So is there a rule as to the cases when the verb can be omitted?
You can never just leave out the infinitive (‘second verb’) in an infinitive construction. If you do, then you are either changing the meaning a tad or you are creating an invalid sentence (or something in-between).
The thing with all your examples (except for *Auto können, I’ll cover that further down) is that the verb can be either a modal verb requiring an infintive or a full verb with a potentially slightly different meaning. Unfortunately, I will have to go through the list one by one to point it out.
Ich will nach Hause.
The verb wollen is very much like the English verb to want: ‘Ich will Schokolade’ directly translates to ‘I want chocolate’. The difference is that the German verb is more flexible and can take a few adverbials of place as well. Saying ’Ich will nach Hause’ does not imply any kind of means of getting there, it can be crawling, walking, driving, skiing, teleporting, flying or whatever. It just signifies that you want to be at home right now, no matter how you’re getting there.
Ich muss dringend zum Arzt.
This question covers the use of müssen as a full verb of movement.
Ich mag nicht in die Schule.
This is basically wollen, version 2.0. Since mögen is a more polite way of saying wollen, what I said up above is valid here, too.
Was soll ich damit?
Here, in fact, the was as a question word is asking for the verb that you ‘left out’. So there is no verb missing, it is merely explicitly being asked for. A similar thing is the case for demonstrative pronouns as in the following example:
Ich darf das nicht.
Here, das is explicitly referencing the verb of whatever it is you are not allowed to do. Adding an explicit ‘placeholder verb’ (like English would with to do — ‘I’m not allowed to do that’) wouldn’t be wrong but less general.
Ich kann Englisch.
Em1 gave an overview of können actually meaning to be proficient (in sth.). Compare ‘Ich kann Stickereien’, meaning that you are proficient in creating embroideries. This is a slightly different meaning than if a verb (’Ich kann Englisch sprechen’) is added, in which case the können would reduce to a typical modal verb.
*Ich kann Auto
This sentence does not work, because car is not something you can be proficient at.
Likewise, every case in itself would require its own discussion. But the general principle is that a second verb (if it is there) is typically required to conserve the meaning.
Technically we would have to discuss each of the verbs separately, but I'll lump them all together.
These modal verbs are not just modal verbs. Some of them have a definition as a full verb, too, and others can be used as full verbs in some circumstances.
Strictly speaking, I would never understand "Ich kann Englisch." as a shortened form of "Ich kann Englisch sprechen.". It's plainly "Englisch können". Können is defined as beherrschen or wissen. As such, können would probably be translated as to know; thus; "Ich kann Englisch." is "I know English." and not "I can speak English.".
Können is kinda a replacement for beherrschen in that example. "Ich beherrsche Englisch." is certainly not a 'good German' sentence, but it's correct.
Anyhow, you can complete the sentence with an infinitive, see also canoonet. And sprechen is the word that usually is being implied when saying "Ich kann Englisch" as opposed to schreiben, lesen etc. You would have to mention those verbs, if you'd like to express another sense other than knowing a language. (Besides, that's the reason why your notion that you can always drop the main verb cannot be true.)
Looking at the other examples, in "Ich will nach Hause.", for example, gehen is not necessarily the implied action. Fahren, laufen, or even hüpfen are valid verbs. However, wollen is defined as beabsichtigen or anstreben. You're just saying that you want to get home, or rather that you would feel much better if you were at home.
"Ich muss zum Arzt." means that it's (more or less) urgent that you visit a doctor. Müssen implies that it is necessary to get somewhere; or away as in "Ich muss weg.".
None of these sentences is really a shortened form; though, a certain verb is often implicitly implied.
So, why doesn't work "Ich kann Auto."? The verb fahren is certainly the most likely verb. What else could you do with a car?
I don't really have a satisfying answer to that outside of mentioning that modal verbs used as full verbs are not completely universal. "Auto können" is kinda nonsense, as is "Auto beherrschen".
Funny enough, if someone asks you "Kannst du Auto fahren?", you're fine answering "Ja, kann ich." because then it's obviously an ellipse of "Ja, ich kann Auto fahren.".
I feel like my answer isn't complete and I hope others can elaborate more on the "why", but hopefully I could make clear that modal verbs are more than just modal verbs.
