Question simplified and clarified as requested... thanks for the feedback! If humankind evolved on a planet with a consistent, gentle environment, would it negate the need for them to develop variation in skin tone, eye shape, nose shape, etc.?
Extreme weather does not really cause evolution into races/species. The cause is reproductive isolation. Isolate groups of your initial population, and they will eventually evolve differences just through random variation, even it the weather &c is the same for all.
Differences in climate can, and often do, determine which variations are selected for, so for instance animals living places with snowy winters may evolve two distinct seasonal color phases, while closely-related ones in warmer climates may stay the same color all year.
I agree with jamesqf.
I would add that IRL we observe such isolations determined by mountains, valleys, canyonds, deserts, rivers, and dense forest; and these do not have to be particularly remarkable: I recall reading of one instance in which monkeys of one ancestral species are evolving differently on two sides of a river that is barely 30 feet wide. Too far to cross by swinging in the trees available, and too fast to risk wading. Presumably some rare drought a few hundred thousand years ago shrank the river enough to cross. Or froze it, or a flood washed a tree full of monkeys across, or whatever.
Such minor obstacles mean nothing to modern humans; but early human ancestors could be equally flummoxed and result in isolation. It isn't that they can't climb the mountain: Perhaps 9999 years out of 10000, there is no good reason to climb the mountain; water and food are plentiful in the grassland and woods fed by the runoff from the mountain, so life is good, barring extreme climate change, and mountain climbing is dangerous and harsh: A 1000 year drought might send you up there, and on the other side you might find another species isolated for 500 generations, with genetic drift altering their physical appearance (different races) and perhaps making them mutually infertile (different species).
In genetic recombination, not all genes are mix-and-match, and many phenotype features depend on many genes working in concert. The concert can be disrupted if some of those genes from one parent do not fit perfectly with those of the other parent: So (A) has 'abcdef' genes, (B) has 'abgdeh' genes; the 'g' and 'h' are alleles of 'c' and 'f', respectively.
So while the combos 'cf' and 'gh' work fine together, and end up producing the same product of two proteins, the combos 'ch' or 'gf' will clash: what is produced by 'c' won't fit with what is produced by 'h', likewise for 'g' and 'f'.
Thus for these parents (A) and (B), half their children are stillborn, or are spontaneously aborted during development, perhaps even at the egg stage.
Variation like this can arise through genetic drift, and enough such variation can make species that look the same infertile. (Consider how many modern human couples have had difficulty conceiving, or consider how a horse and donkey can mate but produce infertile offspring (mule or hinny)).
This is one mechanism that can cause species to emerge: Isolation can immediately begin the process of genetic drift, which is a random walk, eventually this causes "races" (still mutually fertile), longer isolation will eventually cause speciation (mutually infertile). Along that road another town allows mutual fertility with infertile (or genetically flawed) offspring; the Horse+Donkey = (Mule or Hinny) paradigm; i.e. hybrids are infertile, or have some congenital problem that effectively prevents mating (e.g. a horse born congenitally blind, in the wild, may technically be fertile but never survive to mate.)
A planet with a constant environment is nigh on impossible. After several attempts to devise a planet with a constant environment they met with failure. Astronomical factors like axial tilt and the shape of the planet's orbit mitigate against environmental constancy. Ditto with surface features like its geography and whatever diversity it would have in its range of biomes.
This consideration alone removes the possibility of a human species shaped by the selective pressures of its environment in a uniform and unchanging manner.
It is noted that other answers favour genetic drift caused by isolation. While this will play a role, it is not most likely cause of ethnic diversity. [For the purposes of this ethnicity refers exclusively to the phenotypic characteristics of human beings with culture excluded.]
On Earth when humans moved into Europe they lost their darker skin tones in order to synthesize vitamin D. As explained above, there is always likely to be geographical locations where environmental selection pressures will play an active role in shaping ethnicity.
However, the main driver of ethnicity will be sexual selection. Where mates choose their partners on the basis of agreeable characteristics. Some of which will involve selecting partners who will be expected to bear the healthiest offspring.
A planet with a constant environment, assuming such a hypothetical construct could exist, but we can pretend it does, would not prevent ethnic diversity arises within a human species. However, despite their ethnic differences, they remain, like here on planet Earth, one species.