It's not a phrasal verb—or at least, to avoid precise, dispute-prone terminology in a topic where the distinctions are blurry, it doesn't work like pick up. The word apart works like the words in bold face in these sentences:
You look marvelous!1
The breakfast cereal grew soggy.
Jody feels blue.
The cat is on the mat.
In that moment, Bobby became a man.
Boldheart grasped his sword and ran towards the dragon, and at every step he took the monster became smaller and smaller and smaller, until at last, when Boldheart was quite near it and struck at it with his sword, the beast had become so tiny that he had not the heart to kill it.2
The terminology isn't important, but if you want a grammatical term for these, some people call them subject complements, because they name some attribute that the verb asserts of the subject. "Marvelous" describes you, "soggy" describes the breakfast cereal, "blue" describes Jody, etc. And when people grow apart, "apart" describes the people.*
Apart is a kind of condition: most literally, when two things are separated by some distance between them, they're apart. Being apart is the opposite of being together. To grow apart is to become separate. Since it takes two things to be separate, you either need a plural subject, like this:
Over the years, I and my wife grew apart.
Over the years, we grew apart. (That is, we became more and more separate.)
or a singular subject with something after apart that tells what it's apart from:
Over the years, I grew apart from my wife.
or a singular subject with a verb that, together with apart, means that the subject got divided into parts:
The radio-controlled mouse came apart when it crashed into the wall.
To really understand apart, see these excerpts:
The skyscraper rests on 1,200 great pine piles, some of them 40 ft. long, which were driven into the sand of the site. These piles are in rows, two feet apart, under the vertical columns which support the building.3 (That is, there is a distance of two feet between each row and the next row.)
Apart from cartoons, there is a growing and substantial demand by the American Press for all forms of humour.4 (That is, in addition to the demand for cartoons, separate from the demand for cartoons.)
One woman with long, fair, braided hair and a gown more rich than that of the others sat a little apart from the rest.5 (That is, she sat in a position separate from the rest of the people.)
I agree with StoneyB that the usual phrasings for your meaning are grow out of and outgrow (which is like a phrasal verb except that the preposition has been stuck to the verb, as it's done in Russian).
Sources, some slightly edited to shorten or modernize them:
1Billy Crystal / Fernando Lamas.
2"The Magic Flute: A Story for Children" by Amy F. Cackett. The Strand Magazine, Vol. xxii, No. 129, October 1901, p. 349.
3"Building a Skyscraper" by Ray Stannard Baker. The Strand Magazine, Vol. xix, No. 112, April 1900, p. 381.
4"The Humorous Artists of America" by Thomas E. Curtis. The Strand Magazine, Vol. xxiii, No. 135, March 1902, p. 310.
5"The Mutiny on the 'Potemkin'" by A. Kovalenko. The Strand Magazine, Vol. xxx, No. 180, January 1906, p. 617.
*Some people say that "on the mat" is not a subject complement but an adverbial phrase (or something else). This is why the terminology is not important. The important thing I'm pointing out is that in "We grew apart", "apart" describes "we".