Is what the husband did—abandoning his wife due to her not lowering her gaze, her submissive speech with men, and her talking with her colleagues about matters outside of work, and the recurrence of this despite his warnings, his abandonment of her, and her swearing on the Quran—the appropriate action?

A husband must prevent his wife from committing evil deeds and from engaging in suspicious mixing (with men) at work, due to the qawamah (guardianship/authority) that Allah has granted him. Otherwise, she is considered disobedient (nashiz), and she should be divorced if means of reconciliation are not effective with her.