Simply put: no.
The reason is also simple: when Eastern Europeans conquer the Ukraine, they always head straight to Odessa, on the Black Sea, where they stretch out on the beach until the next turn.
Seriously, an attack, whether successful or not, takes place during the Combat Movement (and combat resolution) portion of your turn. A unit can make only one attack per turn. Any units (except air) that executed an attack during the combat movement phase are finished doing anything for the remainder of your turn. (If you choose to retreat any of the units, that retreat occurs during the combat movement-resolution phase, and ends that attack.)
Immediately following the combat movement phase, and the resolution of all of your attacks, is the next portion of your turn: Non-Combat Movement. In this phase, you can move ONLY those units that did NOT participate in the combat movement phase. (The only exception is air units; those which were in combat must now land.) Remember, you make ALL your combat movements first. THEN you resolve all combat. THEN you go into non-combat movement.
There is no case in which a unit can attack into more than one territory during your turn. Well, there is “blitzing,” in which your armor zooms into an EMPTY enemy territory during combat movement, then zooms into an adjacent enemy territory to make a second attack on the same turn. It is perfectly OK to attack empty enemy territory – this is one form of attack during combat. (However, you cannot enter an enemy territory during your non-combat phase, even if it is empty.)
So, if Ukraine was EMPTY and you rolled in unopposed with armor, then your armor could continue right on through to Caucasus. But this is the ONLY case.
What constitutes a turn?
As it says in the manual, your turn consists of these sequences:
1. Dvlp special weapons/purchase units
2. Combat movement
3. Combat (resolution)
4. Non-Combat movement
5. Place purchased units
6. Collect income.
