His driving the car isn't wrong. The book might be wrong. It depends on whether the book is talking about nouns in particular.
We can use a gerund-participle clause in many of the same situations where we use noun phrases. We can use them as Subjects, Objects and as the Complements of prepositions. A gerund-participle clause is a clause which uses an -ing form of the verb with no (present or past tense) auxiliaries. The part in brackets below is a gerund-participle clause:
- I don't like [Bob driving my Ferrari]
Gerund participle clauses can have either normal noun phrases, or genitive ones using 's or s' as Subject:
- [Bob driving my Ferrari] was annoying.
- [Bob's driving my Ferrari] was annoying
If the Subject of the gerund-participle clause is a pronoun, we can use an accusative one or a genitive one:
- [him driving my Ferrari] was annoying.
- [his driving my Ferrari] was annoying.
Notice that gerund-participles are verbs not nouns. Because of this, they take Direct Objects. We use adverbs and not adjectives to modify the -ing verb:
- Him carefully driving my Ferrari round the race-course annoyed me.
- His carefully driving my Ferrari round the race-course annoyed me.
In both sentences above we see the Direct Object my Ferrari and the adverb carefully. Notice that if we use the adjective careful to modify the verb the sentence will be ungrammatical:
- *Him careful driving my Ferrari round the race-course annoyed me. (ungrammatical)
- *His careful driving my Ferrari round the race-course annoyed me. (ungrammatical)
Many nouns in English describe actions:
- inspection, theft, release, baptism, massacre, emission, liberation
Nouns do not take Direct Objects. If we want to show the people or things that these actions were done to, we need to use a preposition phrase using the preposition of:
- the inspection of the school
- the theft of the jewels
- the release of the prisoners
- the baptism of the congregation
- the massacre of the Daleks
- the emission of greenhouse gases
- the liberation of the prisoners
If we want to modify nouns, we need to use adjectives, not adverbs:
- the careless emission of greenhouse gasses
- *the carelessly emission of greenhouse gasses (ungrammatical)
Noun phrases in English often have Determiners. These are words like a or the that do a special job in noun phrases. We can use genitive noun phrases using 's or s' as Determiners. We cannot use plain form ones:
- Tom's theft of the jewels
- Tom theft of the jewels (ungrammatical)
We can use genitive pronouns as Determiners. We cannot use accusative or nominative ones:
- His theft of the jewels
- *Him theft of the jewels
- *He theft of the jewels
Of course we can always use determiners like the:
Nouns ending in -ing
In English we can use different endings to turn verbs into nouns. For example we can add -(a)tion on to many verbs to make a new noun:
- evaporate, evaporation
- cite, citation
- inspire, inspiration
We can also make new nouns from almost any verb by adding -ing to the plain form of the verb:
- run, running
- read, reading
These look exactly the same as gerund-participles, but they are not verbs. Like all nouns, they take preposition phrases with of instead of Objects:
- The running of the race
- The reading of the rules
- *The running the race
- *The reading the book
Like other nouns they can take genitive noun phrases and prepositions as Determiners. They cannot take normal nouns or accusative or nominative pronouns as Determiners:
- Tom's running of the race
- His running of the race
- *Tom running of the race (wrong)
- *The man running of the race (wrong)
- *Him running of the race (wrong)
- *He running of the race (wrong)
Of course, we can always use the as a Determiner:
Like other nouns, these nouns can be modified by adjectives. They can't be modified by adverbs:
- His careful running of the race
- *His carefully running of the race (wrong)
If we use these noun -ing forms as nouns, then they are nouns, not verbs!
The Original Poster's question
- I don't approve of his driving of the car.
- I don't approve of his driving the car.
The first sentence uses the noun driving. It has a preposition phrase to show what he was driving. The second sentence uses the gerund-participle form of the verb, driving. Because it's a verb it takes a Direct Object.
Notice that we can use the instead of his as a Determiner in (1). Also we cannot use the accusative him as a Determiner:
- I don't approve of the driving of the car
- *I don't approve of him driving of the car (wrong).
Notice that we can use him as a Subject in (2). Obviously we can never use the as the Subject of a verb;
- I don't approve of him driving the car.
- *I don't approve of the driving the car. (wrong)
Lastly, notice that we can use an adjective as a modifier in (1) because driving is a noun. We can't use an adverb. In (2) where driving is a gerund-participle verb, we can use an adverb as a modifier but not an adjective:
(1) I don't approve of his careless driving of the car. (noun with adjective)
(1) *I don't approve of his carelessly driving of the car. (noun with adverb, wrong)
(2) I don't approve of his carelessly driving the car. (verb with adverb)
(2) *I don't approve of his careless driving the car. (verb with adjective, wrong)
So, in the Original Poster's example, driving could be a verb or a noun. If it is a noun, it takes a preposition phrase, of the car. If it is a verb it can take a Direct Object, the car. If driving is a noun, the word his is a Determiner. If driving is a verb, we probably want to think of his as a Subject.
Hope this is helpful!
All of this information can be found in the Cambridge Grammar of the English Language, the famous reference grammar by Huddleston & Pullum (et al) 2002. You can also find most of it in A Student's Introduction to English Grammar, Huddleston & Pullum, (2005)