一、that day 本身可用作状语,意为“那天”,其前通常无需用介词 on.如：
He was ill that day.那天他生病.
We weren’t in town that day.那天我们不在城里.
He asked us whether we were going away that day.他问我们是否那一天走.
She wasn’t feeling well that day.那天她感到不太舒服.
We were less than delighted to have company that day.那天有客人我们并不太高兴.
She didn’t phone that day,nor the next day.她那天没有打电话,第二天也没打.
About 30 percent of the pupils were absent that day.那天大约有30%的学生迟到.
二、但是,说 on that day 也不算错,只是不如 that day 常见,如下面一句就出自《朗文当代英语词典》：
What really happened on that day so long ago?
三、对于 that day 与 on that day 的区别,可作如下解释：
1.that day 比 on that day 更普通、更口语化.
2.that day 更常用于美国英语中,就如 He came on Monday 去掉介词on 也多用于美国英语一样.
3.on that day 比 that day 表示的时间概念更明确.