As @Larnu mentioned, use an AFTER UPDATE trigger. However, the issue you have is that without a unique identifier on the table, your update may not be as targeted as you want. You can add some kind of unique identifier field.
CREATE TABLE important_action (
    [uid] int IDENTITY(1,1),
    myAction VARCHAR(50),
    actionStatus VARCHAR(50),
    [date_modified] datetime DEFAULT CURRENT_TIMESTAMP 
);
GO
CREATE TRIGGER important_action_update_date_modified ON important_action
AFTER UPDATE
AS
BEGIN
    SET NOCOUNT ON;
    UPDATE important_action 
    SET date_modified = GETDATE()
    FROM important_action as ia
        INNER JOIN inserted i 
            ON i.[uid] = ia.[uid]
    ;
END
GO
INSERT INTO important_action (myAction, actionStatus)
VALUES ('testAction1', 'ts1')
    , ('testAction2', 'ts2')
SELECT * FROM important_action;
UPDATE important_action
SET actionStatus = 'us2'
WHERE myAction = 'testAction2';
SELECT * FROM important_action;
Results:
First Select:
| uid | myAction | actionStatus | date_modified | 
| 2 | testAction1 | ts1 | 2022-11-02 12:35:45.740 | 
| 3 | testAction2 | ts2 | 2022-11-02 12:35:45.740 | 
 
Second Select:
| uid | myAction | actionStatus | date_modified | 
| 2 | testAction1 | ts1 | 2022-11-02 12:35:45.740 | 
| 3 | testAction2 | us2 | 2022-11-02 12:35:45.757 |