The solarian weapon states that it is supernatural which means it is magical. Does that mean the damage counts as magical for things like incorporeal creatures?
Yes, they do...
It is in the Supernatural ability very definition:
Supernatural Abilities (Su)
Supernatural abilities are magical but not spell-like. Supernatural abilities are not subject to spell resistance and do not function in areas where magic is suppressed or negated (such as an antimagic field). A supernatural ability’s effect cannot be dispelled and is not subject to counterspells.
This point is further confirmed by the definition of incorporeal creatures:
Incorporeal (Ex) An incorporeal creature has no physical body. It can be harmed only by other incorporeal creatures, magic weapons or creatures that strike as magic weapons, and spells, spell-like abilities, or supernatural abilities.
Looking at the Ethereal condition:
An ethereal creature is invisible, insubstantial, and capable of moving in any direction, even up or down, albeit at half normal speed. As an insubstantial creature, you can move through solid objects, including living creatures. An ethereal creature can see and hear on the Material Plane, but everything looks gray and ephemeral. Sight and hearing onto the Material Plane are limited to 60 feet.
Force effects and abjurations affect an ethereal creature normally. Their effects extend onto the Ethereal Plane from the Material Plane, but not vice versa. An ethereal creature can’t attack material creatures, and spells you cast while ethereal affect only other ethereal things. Certain material creatures or objects have attacks or effects that work on the Ethereal Plane.
An ethereal creature exist on the Ethereal plane and follow the same rules than any other planes about interacting with the Material Plane, with the exceptions above. So wielding a magical sword on the Material Plane do not bother a creature on the Ethereal Plane more than a creature on the Fire Plane, except if this weapon is enchanted with an appropriate Force or Abjuration effect. I think you were referring to Incorporeal creatures, as mentionned above.
I believe solarian weapons do not count as magical for overcoming incorporeal damage unless the solarian has a weapon crystal set in their mote.
See page 170 of the Core Rulebook, first complete paragraph of the second column:
When a weapon crystal is within a mote and the solarian uses the mote in solar weapon form, the solar weapon is considered magic for the purpose of overcoming damage reduction.
That strongly implies that if a weapon crystal is not in a mote, the solar weapon is not considered magic. There is the detail of "for the purpose of overcoming magic damage reduction," and one could argue that incorporeal immunity is not damage reduction because DR is a specific thing... but honestly, I find it more likely that the intent is "weapon crystal adds magic" the way adding a fusion adds magic to normal weapons.
Regarding the weapon being supernatural, I don't see it called supernatural anywhere. From page 102 of the Core Rulebook:
Your solar weapon functions as a one-handed kinetic advanced melee weapon, and you’re automatically proficient with it. At 1st level, choose whether your solar weapon deals bludgeoning, piercing, or slashing damage. You can change the damage type each time you gain a new solarian level. Your solar weapon deals damage equal to 1d6 + your Strength modifier. This damage increases by 1d6 at 6th level, 9th level, 12th level, and every level thereafter. Solarian weapon crystals (see page 170) can increase your solar weapon’s damage.
That' the full weapon mechanic, and it's not described as supernatural or magical there. The class ability that creates it, Solar Manifestation, is an SU ability, yes--but as written, it seems to supernaturally create a non-magical item.