Learn English – What do “resolved” and “improved” mean when they are used about recovering from a disease

This is a standard idiom in English. The earth here is a metaphor that symbolizes the impossible, the immoderate, or the unfulfillable:

promise (someone) the earth (or moon)^NOAD
make extravagant promises to someone that are unlikely to be fulfilled:
interactive technology titillates, promises the earth, but delivers nothing

The origin of this idiom is probably from the third temptation of Jesus (in the Bible), where the devil tempts him with the ridiculous promise of all the kingdoms of the world:

Again, the devil taketh him up into an exceeding high mountain, and sheweth him all the kingdoms of the world, and the glory of them; And saith unto him, All these things will I give thee, if thou wilt fall down and worship me.
_{— Matthew 4:8–9 (The Bible, King James Version)}

Merriam-Webster's Dictionary of Synonyms, revised edition (1984) offers an extended discussion of the similar word pair retrogressive and regressive, both of which it treats as members of a group of synonyms that also includes retrograde and backward. Here is the relevant coverage:

backward, retrograde, retrogressive, regressive all involve the idea of not moving or going ahead, or forward, or in advance. ... Retrogressive implies opposition to progressive. Like retrograde, and unlike backward, it implies movement in the direction that is the reverse of forward; unlike retrograde, however, it is seldom applied to physical movement; thus, one speaks of a retrograde (but not a retrogressive) movement or rotation, but on might speak of retrogressive (or retrograde) cruelties or behavior when stressing decline from some higher or more progressive level. Retrogressive is sometimes preferred as a milder term when the reverse of improvement or betterment rather than positive decline from an improved or better state is implied {a retrogressive policy} {objections were made to the proposed legislation on the ground of its probable retrogressive effect} Regressive carries a stronger implication of going backward by steps or degrees and often, also, a weaker implication of failure to progress or move ahead than any of the others. Consequently it is often the preferred term when a colorless or uncolored statement of fact is intended; thus, when one infers a cause from an effect or a principle from a number of facts he follows a regressive process of reasoning; the process of growing old may be described as a retrogressive development when the stress is on the absence of progress, and a regressive development when the intent is to indicate that it is marked by an inversion of order in its stages; a regressive loss of memory implies that the most recent memories disappear first and the earliest linger longest.

Applying the general tenor off the distinctions between the adjective forms retrogressive and regressive to the verb forms retrogress and regress, we might say that although both involve figurative (that is, not actual physical) movement in the opposite direction from the one in which progress lies, to retrogress is to move backward (figuratively speaking) to an unspecified extent while to regress is to move backward (figuratively) in a series of incremental steps, stages, or degrees.

I must say, though, that I have no confidence that English speakers in general use these two very similar words with any such distinction in mind.

It also bears noting that retrogress is a far less common word than regress, as this Ngram chart of retrogress (blue line) versus regress (red line) versus retrogressive (green line) versus regressive (yellow line) for the period 1800–2008 indicates:

Between 1850 and 1920 or so, the frequencies of regress, regressive, and retrogressive in published writing seem to have been fairly close, after which regress and (especially) regressive took off; but retrogress has been the least common of the four forms since at least 1840, and in recent decades it has become very rare indeed.