Yyyyyep. He definitely could, and, if anything, that short line actually understates the power that the Roman father had over his family, including his sons. The Romans were notably harsh when it came to the family, and while these punishments were not exactly common, we do have evidence that they did actually happen, from fathers killing their sons to fathers selling their sons into slavery, to fathers disowning their sons, you name it. Those fathers were generally considered to have lost control over their sons, or to have done a shitty job teaching them in the first place, but they were still explicitly granted that power by Roman law. The Twelve Tables, of which we only have glimpses and fragments, discusses parental law on the fourth Table:

Table IV. Paternal Power

• A notably deformed child shall be killed immediately.

• To a father [...] shall be given over a son the power of life and death.

• If a father thrice surrenders a son for sale the son shall be free from the father.

Note that last one. The father could sell his son into slavery up to three times, and you know for laws, there's generally a precedent. They took absolute power seriously. Below is another excerpt, this one from Dionysius of Halicarnassus (who lived during precisely the time period you're curious about) detailing exactly what a Roman father could do to his children, if he so wished. It's a long quote, and much of it merely restates what I've noted (you also have to remember that he's writing to a Greek audience, so there's a bit of shock and awe going on here), but it's fascinating nonetheless.

But the lawgiver of the Romans gave virtually full power to the father over his son, even during his whole life, whether he thought proper to imprison him, to scourge him, to put him in chains and keep him at work in the fields, or to put him to death, and this even though the son were already engaged in public affairs, though he were numbered among the highest magistrates, and though he were celebrated for his zeal for the commonwealth.

Indeed, in virtue of this law men of distinction, while delivering speeches from the rostra hostile to the senate and pleasing to the people, have been dragged down from thence and carried away by their fathers to undergo such punishment as these thought fit; and while they were being led away through the Forum, none present, neither consul, tribune, nor the very populace, which was flattered by them and thought all power inferior to its own, could rescue them.

I forbear to mention how many brave men, urged by their valour and zeal to proof some noble deed that their fathers had not ordered, have been put to death by those very fathers, as is related of Manlius Torquatus40 and many others. But concerning them I shall speak in the proper place.

And not even at this point did the Roman lawgiver stop in giving the father power over the son, but he even allowed him to sell his son, without concerning himself whether this permission might be regarded as cruel and harsher than was compatible with a natural affection. And, — a thing which anyone who has been educated in the lax manners of the Greeks may wonder at above all things and look upon as harsh and tyrannical, — he even gave leave to the father to make a profit by selling his son as often as three times, thereby giving greater power to the father over his son than to the master over his slaves.

For a slave who has once been sold and has later obtained his liberty is his own master ever after, but a son who had once been sold by his father, if he became free, came again under his father's power, and if he was a second time sold and a second time freed, he was still, as at first, his father's slave; but after the third sale he was freed from his father.

This law, whether written or unwritten, — I cannot say positively which, — the kings observed in the beginning, looking upon it as the best of all laws; and after the overthrow of the monarchy, when the Romans first decided to expose in the Forum for the consideration of the whole body of citizens all their ancestral customs and laws, together with those introduced from abroad, to the end that the rights of the people might not be changed as often as the powers of the magistrates, the decemvirs, who were authorized by the people to collect and transcribe the laws, recorded it among the rest, and it now stands on the fourth of the Twelve Tables, as they are called, which they then set up in the Forum.

And that the decemvirs, who were appointed after three hundred years to transcribe these laws, did not first introduce this law among the Romans, but that, finding it long before in use, they dared not repeal it, I infer from many other considerations and particularly from the laws of Numa Pompilius, the successor of Romulus, among which there is recorded the following: "If a father gives his son leave to marry a woman who by the laws is to be the sharer of his sacred rites and possessions, he shall no longer have the power of selling his son." Now he would never have written this unless the father had by all former laws been allowed to sell his sons.

EDIT: Because I'm bored, and because corroborating sources are always delightful, here are a couple of points from Quintilian's letters, too! :D Note: Feel free to flip through them if you like reading someone discussing semantics about Roman law. It's exactly what you would expect.

A father who recognises a son whom he had exposed in infancy, shall only take him back after paying for his keep.

[...]

A disobedient son may be disinherited.

[...]

The man who refuses to appear in defence of his father when accused of treason shall be disinherited: the man who is condemned for treason shall be banished together with his advocate.

And fun set of little semantic arguments here...

Take the following case as an example. "A father shall be empowered to arrest his son, and a patron to arrest his freedman. Freedmen shall be transferred to their patron's heir. A certain man appointed the son of a freedman as his heir. The son of the freedman and the freedman himself both claim the right to arrest the other." Here the father claims his right over the son, while the son, in virtue of his new position as patron, denies that his father possessed the rights of a father, because he was in the power of his patron.

[...]

The man who kills his father shall be sewn up in a sack. What if he killed his mother?

[...]

It is illegal to drag a man from his own house into the court. What if the man was dragged from his tent?