If someone marches and uses all of the values from his army in a fight with another house, can he leave some of those units behind, or does he have to move everything he fought with?

Everything you fight with must move into the zone if you win. Additionally if you are moving all units you must decide before the combat if you leave a power token or not. If you want to leave anyone behind they may not participate in the battle.

Daices said:

It helps if you realize that a March order is a "Move" order. The units actually move into the other territory and fight there, not "on the border". If a unit moves into another territory, it cannot also be "left behind" in its starting territory.

kauai1964 said:

Wouldn't it be helpful if this aspect of combat were specifically stated in the rules or errata instead of only being implied? FFG, can you give us a ruling?

Please re-read the Rules:

March Orders are resolved during the second step of the Action
Phase. When resolved, all units in the Order token’s assigned area
may march (i.e., move) to one or more adjacent areas. If marching
units enter an area containing one or more enemy units, a combat
ensues.

When resolving a March Order, the following rules apply:

A player may move all, some, or none of his units from the
area assigned the March Order

If your units move "from" and "into", how can they be left behind?

--

Also, under Retreats:

If the attacker lost the combat, his surviving units must retreat
back to the area from which they marched.

How could units retreat and be routed if they were left behind in the first place?

--

Then under Establishing Contol:

When a House vacates a land area (i.e., all its units leave during
a March Order) the House may elect to establish control by placing
a Power token from its available power in the vacated area.

It says the units leave during the March Order - not after the combat.