The Koreans were still a Chinese tributary state at this point. I don't think the Japanese would want to do this without being confident that they could defeat a Chinese army and local Korean resistance, and I wouldn't assume such confidence is warranted this early.
I would not assume that Japan would defeat an intervening Chinese army either.
However I would not say that fear of China's potential was a guaranteed deterrent for the Japanese Emperor, Saigo and friends. The Japanese side (at least absent the
Genro personalities who were on a world tour) could have the confidence to *try* an invasion, whether it actually turns out well for them or not.
Japan at around this time was already willing to squeeze out Chinese claims of suzerainty over the Ryukyu islands, and was willing to invade Chinese own territory in Taiwan, as early as 1874.
As for what the Chinese would do in reaction to a Japanese invasion and attempted occupation of Korea in 1873, that is a very interesting question. On the one hand, the Chinese of the 1880s contested Japanese moves in Korea, willing to escalate to war by 1894. The Chinese of 1873 had also defeated the Taiping and Nian rebellions in the heart of China proper. They could be in a better position to defend their interests in 1873 than in the previous two decades.
On the other hand, the Chinese were still dealing with the Muslims rebellions in the southwest and northwest, so Viceroy Li Honzhang in Zhili would know that General Zuo Zongtang would not be available to reinforce him and General Zeng Guofan had died the prior year. In 1876, the Chinese did nothing to counter Japan's gunboat diplomacy to impose an unequal treaty on Korea (treaty ports and extraterritoriality).
Whether or not the Chinese send an army to help Korea, one consequence of a precocious Japanese invasion of Korea in '73, especially a successful occupation, would probably be to officially open Manchuria to Han Chinese settlement a couple decades ahead of OTL. Of course, unofficial settlement of Han had begun over a hundred years earlier despite repeated attempts to ban it.
A Japan defeated in Korea, resting from successful exertions there, or bogged down in pacification would likely have a poorer bargaining position vis-a-vis Russia by around 1875, and may have to yield Sakhalin to Russia, without getting the Kurils as compensation, as occurred in OTL:
https://en.wikipedia.org/wiki/Treaty_of_Saint_Petersburg_(1875)