Korea and Japan seem to have a week system that looks like this:
Monday 月曜日
Tuesday 火曜日
Wednesday 水曜日
Thursday 木曜日
Friday 金曜日
Saturday 土曜日
Sunday 日曜日
China apparently used to be the same as well.
星期 wikipedia:
老黄历的值日星君就是曜日。日曜日是星期日，月曜日是星期一，火曜日是星期二，水曜日是星期三，木曜日是星期四，金曜日是星期五，土曜日是星期六。中国在民国成立后改称星期，其中的“星”字便是指这七曜，但在日本、韩国和朝鲜仍沿用曜日名称。台湾日治时期由日本引进星期制，因此亦使用七曜，但战后由中华民国统治后改成星期。
The answer to this question is quite complicated, there is a whole website devoted to this subject here.
Shrinking things down as short as possible, the idea of seven day periods within a calendar based primarily on months came from the West into China, possibly more than once. During the Tang dynasty, it was taken up in China for use in astrology. For the percentage of the population who were not astrologers, it was not an important or widely known idea, and by the Ming dynasty, it was simply an odd factoid in astrological treatises.
According to the same website, the Japanese got the idea of the seven day period through Buddhist sources in the early 800s. Both the Chinese and Japanese thus got this idea, either directly or indirectly, through Western Buddhist or Manichaean sources, which identified the days with the sun, moon, and five planets, the 'seven luminaries' 七曜. Unlike China, the astrological use remained popular in Japan, but was limited to astrology until Japan decided to adopt the Western system of weeks as part of its calendar reform in 1876.
The seven-luminaries names of the seven days thus continued to be used in Japan because they had a continuous history. In China, the whole thing was forgotten until they were reminded of the idea when Westerners starting setting up extra-territorial concession across China in the second half of the 1800s. The concessions naturally used weeks to regulate both work schedules and religious worship, particularly the Sabbath.
It seems clear that libai 禮拜 'worship' as the name for a week originated some time in this period; there is a very interesting article by 張書華 called 西制東漸：試探“星期制”於清末上海之傳布 which describes the spread of the week throughout China, a spread in which Shanghai played a leading role.
Yuan Chia-ku 袁嘉谷 was the Qing official who was supposedly responsible for coining the name xingqi 星期 'a stellar period' for the week in the last years of the Qing dynasty, but it was not until the Republic of China was declared that the Western calendar was adopted and the 'week' became a part of Chinese time-keeping. Even then, the use of libai remained much more common than xingqi, until the dawn of the People's Republic in 1949.
I have to say that Yuan's objection to the use of the Tang era names for the days of the week doesn't seem that unreasonable to me. If we had to go through a major calendrical reform (heaven forfend), would you like to have a new system of days set up named after the Buddhas of the seven heavens?
According to this article, in 1905, 袁嘉谷 was ordered by Emperor 光绪 to head a division meant to compile and translate textbooks in a standard way. As part of this, they encountered a problem with things not having a consistent name; one of these was the name for the unit of one week.
Although the Chinese 曜日 did happen to correspond to the Western idea of one week, 袁嘉谷 and his colleagues thought it was annoying to say and thus changed it to 星期:
但袁嘉谷感到不順口，使用起來不方便，與同事們商量後，將一週稱一星期，以“星期日、星期一……星期六”依次指稱周內各日。
History reason.
The history of China using *曜日 system could track back around AD 300ish. This system is not widely used by ordinary people.
It's Korea and Japan followed up with China because ancient Korea was a vassal state of ancient China most of the time, and ancient Japan are regarded less civilized then ancient China, thus, a lot of traditions of ancient China are preserved well in Japan and Korea.
And in late Qing dynasty, China started learned from western world so rename the system(or in other words, format the system and name it "星期" rather than "曜日"
