Remote rest api supports selected methods, as per my understanding we dont have method to check task completion date. But you can print or set date& time into process variable from 'OnExit' script of the human task. Hope it helps..
Were you able to use task event table to get task completion date or did you used on exit event/script?
I found the solution myself. Hope this can help someone..
List<TaskEvent> taskEvents = taskService.execute(new GetAuditEventsCommand(taskId, new QueryFilterImpl(0,0)));
You can add filters for task status. Also, you can use default constructor of GetAuditEventsCommand if you want all the task events.