So, the basic response is, "of course not". The complicated response is: "sort-of".

Like many regions on the fringes of Han China (meaning the area of China dominated by Han ethnicity), the Korean peninsula was under administrative control of Imperial China at some points in history. Usually, this meant some kind of garrison, along with a regional diplomat. So, lets look at some points in history in which the peninsula was partially controlled by the Chinese state:

Han Dynasty ~200 BCE - ~200 CE - there is significant evidence (textual, steles) that the Chinese maintained military/official presence in the north, but not so much in the South. The commanderies set up to govern Korea lasted to some extent for the whole of the Han period.

Yuan Dynasty - The Khans really wanted Korea, and Japan. The Japan story is well known. Korea, of course is often overlooked. The war with Korea lasted around 70 years (dwarfing the two invasions of Japan in scope and consequence). By about 1280, the Koreans capitulated and became a vassal state, a relationship that lasted through the end of the Yuan period in the 1360's.

So, how do we interpret Trump's statement?
Not terribly defensible. The problem is that even in the Han, when there was official presence, we are talking about more of an occupation than a full incorporation. Surely there was sinification, which extended to comparatively never occupied Japan also in the Tang, but this does not mean we are talking about any kind of demographic effects that might suggest a joining of "Korea" and "China", like those demographic shifts that happened everywhere in what we now call China (save Tibet before the late 20th Century CE). There was no large population movements which led to Han mixing with non-han peoples.
These shifts happened in the Tang, Yuan, Ming, Qing, and in the PRC, and can be argued as being expansions of "China".

Again, none of this can be said about the Korean Peninsula. As much as Xi would like it to be so, or Trump would like to parrot it because he is a tactless diplomat.

The point to note for the Han commanderies is that what is Korean and what is not at that time period is defined by identities rather than territory due to the lack of an actual Korean state. While "China" may have owned land on the Korean peninsula, neither Koreans today or the states that preceded the current two republics claim descent from that said occupation. Once the Han dynasty started to disintegrate, the Han commanderies were quickly subsumed into Goguryeo and neither Goguryeo or Balhae ever saw themselves as Chinese or as part of China, never mind the southern portions of the peninsula.

The Yuan dynasty is a considerably different situation in that despite the initial long periods of resistance, Goryeo became heavily integrated into the Yuan. In addition to the territories that Yuan took from Goryeo to set up separate commanderies, it also heavily interfered in Goryeo's internal affairs, replacing multiple kings as well as setting up directly governed ministries/departments for the sake of collecting tribute. It was never fully integrated however, and the fact that these commanderies were set up, and the presence of the special deparments also means that Goryeo was normally not treated as a direct part of Yuan, and that its territories and governance were kept separate, retaining a considerable amount of autonomy compared to other lands ruled by the Yuan. It is the interpretation of the degree of autonomy as to whether to treat it as separate, or as part of Yuan during this period.

But even then, the speed at which the Goryeo dynasty rid itself of its Mongol influences and began to act against the Yuan once Gongmin of Goryeo rose to the throne, as well as the early conflicts and subsequent tributuary status between the two's successor states of Ming and Joseon, shows that while Goryeo may have been ruled by the Yuan, it never developed a greater Mongol or Chinese identity, and neither was it a part of the concept of "China", even accounting for the constant expansions of the concept.

Even modern Chinese historiography do not directly put the Koreans under the umbrella of China, although do use Goguryeo and its ties to Manchuria to claim Goguryeo's, and thus part of Korea's, history as part of its own. This is a very real dispute between Chinese and South Koreans that is ongoing even now, though I believe details on that would violate the 20 year rule.

I'm not quite sure what to make of this thread. Would those who have answered please provide sources? Thanks!

Edit: Thanks /u/bbyeok and /u/dukeofcai.