This dates from 3.5e (or earlier?) - the quote below is from the d20 SRD, and is the same in the PF SRD.
Double Weapons
Dire flails, dwarven urgroshes, gnome hooked hammers, orc double axes, quarterstaffs, and two-bladed swords are double weapons. A character can fight with both ends of a double weapon as if fighting with two weapons, but he or she incurs all the normal attack penalties associated with two-weapon combat, just as though the character were wielding a one-handed weapon and a light weapon.
The character can also choose to use a double weapon two handed, attacking with only one end of it. A creature wielding a double weapon in one hand can’t use it as a double weapon—only one end of the weapon can be used in any given round.
This is open to interpretation, but my opinion is that, as double weapons are already a special case that differ from the general rules for two-handed weapons, that WotC intended for them to be wieldable one-handed as this implies, as well as the explicitly stated two-handed or two-weapon methods.
However...
It appears that Paizo took a different view. There is a feat to specifically allow a character to wield a quarterstaff (a double weapon) one handed.
Quarterstaff Master (Combat)
You can wield a quarterstaff as either a two-handed or one-handed weapon.
Prerequisites: Weapon Focus (quarterstaff ), base attack bonus +5.
Benefit: By employing a number of different stances and techniques, you can wield a quarterstaff as a one-handed weapon. At the start of your turn, you decide whether or not you are going to wield the quarterstaff as a one-handed or two-handed weapon. When you wield it as a one-handed weapon, your other hand is free, and you cannot use the staff as a double weapon. You can take the feat Weapon Specialization in the quarterstaff even if you have no levels in fighter.
While the value of the feat is argued (due to the "WotC" interpretation of the double weapon rules), Paizo's official position is that you can't wield one handed without it.
Hey there folks,
You cannot normally use a double weapon in one hand unless it is sized smaller than you. This feat allows you to get around that restriction.
Jason Bulmahn
Lead Designer
Paizo Publishing
You can wield & attack with two weapons without any special class feature or feat.
Like most contests, however, some restrictions may apply. From the PHB, p195 (emphasis mine):
Two-Weapon Fighting
When you take the Attack action and attack with a light melee weapon that you're holding in one hand, you can use a bonus action to attack with a different light melee weapon that you're holding in the other hand. You don't add your ability modifier to the damage of the bonus attack, unless that modifier is negative.
As long as both weapons are light weapons, you can attack with both at the cost of an action for the first attack, as normal, and then a bonus action for the second attack. The second attack does not receive your ability modifier as a bonus to damage the way attacks normally would.
Fighters & some other classes have an option for their Fighting Style class feature which is also called, confusingly, Two-Weapon Fighting. This feature is not required to attack with two weapons as described above; the benefit it gives is to allow you to add your ability modifier to the second attack's damage.
The Dual Wielder feat is likewise not required to attack with two weapons as described above; it instead provides additional benefits and removes other restrictions/penalties when you choose to do so.
Dual Wielder and the Two-Weapon Fighting style make you better at fighting with a weapon in each hand, but are not required to do so.
Best Answer
No.
The Double special weapon quality allows you to fight as if fighting with two weapons (a one-handed weapon and a light weapon). Since you can't use two weapons in one hand, you can't use a double weapon as a double weapon with one hand.
Your scenario is essentially the same as dual-wielding small quarterstaffs (with a size penalty, of course). You're still limited to one attack each with your main- and off-hand weapons.
Using a double weapon in each hand lets you make one attack with each; using a double weapon with Spear Dancer Style lets you make two attacks with one weapon.