The source for this is Galen's On exercise with a small ball (5.899-910 ed. Kühn). A translation is available in the Oxford World's Classics edition of Galen: selected works (tr. P. N. Singer). He doesn't actually specify a game: it's more that, as he puts it in his opening discussion, many other forms of exercise -- like hunting -- require considerable wealth or free time, while playing with a ball is open to anyone in almost any situation.

We know the names of several ball games from Mediterranean antiquity, but they're also from a bunch of different time periods, and different places. Trigon, for example, seems to have been a particularly Roman game; keretisdein and ourania are too early. The New Pauly s.v. 'Ball games' gives a run-down on these. There are various others mentioned, like a form of keepy uppy where you bounce a ball against a wall (rather than up in the air).

Balls themselves appear to have been usually made of leather and filled with hair, air, or feathers.

The most popular candidate for a particular sport that he meant is one known as harpaston or harpastum, from the word for the ball: Greek harpaston means 'the thing that you're supposed to snatch'. This would be one of the ones filled with hair, so a relatively heavy ball. Galen doesn't call it that -- he just uses the generic word sphaira -- but a couple of references in his discussion tend to support that he's thinking of harpaston or something very similar.

Harpaston was a Roman-era game that is attested in a few sources. Here's what the New Pauly encyclopaedia has to say:

Harpaston ... Name for a small, firm ball, then also for a catch ball game using such a ball (Poll. 9,105; Ath. 1,14f.), similar to the phainínda (cf. Clem. Al. 3,10,50 [and schol.]). The harpaston was a very physical combat game; details of the game are not known. One party attacks the player of the other side, who is in possession of the ball, and attempts to wrest the ball from him (ἁρπάζειν; harpázein, ‘[hastily] grasp’, ‘snatch’, ‘rob’). This player strives to pass the ball to his team mates who in turn are prevented by the attackers from catching the ball. The side in possession of the ball tries to dominate the middle part of the playing field, while the opposing side tries to gain possession of the ball and themselves penetrate the middle part of the field. This resulted in obstructions because of pushing, tripping up or kicking and wrestling over the ball, in the course of which apparently all actions were permissible (Sen. Ep. 80,1-3; Epict. 2,5,16; Mart. 4,19,6; 7,32,10; 14,48). The harpaston required a high degree of skill and physical agility.

These references seem to be all that is known about the game. It seems a decent match for what Galen says about ball sport at 902 Kühn, starting at line 14:

When, for example, people face each other, vigorously attempting to prevent each other from taking the space between, this exercise is a very heavy, vigorous one, involving much use of the hold by the neck, and many wrestling holds. And so the head and neck are exerted by these holds by the neck, and the lungs, chest, and stomach by the laying hold, pushing away, and levering involved in such clinches, as well as the other wrestling­style holds.

On the other hand, Galen insists that ball sport is safer than other forms of exercise, and this is much harder to reconcile with the New Pauly account of harpaston. 909 Kühn:

I should not like to omit from my account of the positive attributes of this exercise the fact that it is free from risk. This is not true of most other sorts of exercise. Running, for example, has frequently killed people, by the rupture of a vital vessel; ... Vigorous horse­riding can cause ruptures in the region of the kidneys, as well as damage in the chest area, or even sometimes in the spermatic channels... The jump, the discus, and the exercise involving turning have also caused many injuries.

Singer's notes on Galen's essay just say,

the exact nature of this sport is not clear; but it is interesting to see G.'s advocacy of a particular kind of physical training, in view of his opposition to the 'athletic' type ...

Galen seems clear that his preferred game involved both wrestling moves, and safety from injuries. To me it sounds contradictory. So who knows.