After the first day of fighting at Gettysburg the union soldiers managed to retreat to the high ground surrounding the city and began to dug in. Why did Lee think it was important to continue the fight?
The fighting was taking place in the north so it’s not like he had to worry about defending southern towns from capture. He could have left and either marched on the factories of Philadelphia or even gone south the Washington. The raid on the north was less about attacking the union army as it was destroying the North’s willingness to continue fighting.
I guess the biggest point is that one of the most important rules of warfare is don’t let the enemy choose your battlefield. So once he saw that the union had the advantage of position why not leave and either find a place to dig in to force the northern soldiers to attack his trenches. Or go on a Sherman style march to the sea and burn the north, and its industrial advantage, to the ground.
He couldn’t have gone south to Washington because the Union army was in his way, and he couldn’t have gone on to Philadelphia because that would have left the Union army at his back, ready to cut off his lines of supply.
As a rule, the attacker doesn’t get to play the waiting game. Every day Lee spent in Pennsylvania, his army grew weaker and the Union army grew stronger. He had to resolve the issue, one way or another.
This latter point is important. You don’t leave a large enemy force to pursue you. If you’re moving against a major city, you don’t just cruise up and yell “Game Over!”. You have to defeat its defenses and move into the city to do what damage you intend to do, or seize what you intend to take.
The delay means your pursuer catches up to you and you change from the besieger to the besieged, just like the Germans at Stalingrad.
Once Lee engaged the Army of the Potomac, he was obligated to break it or slink away back to Virginia with pointless losses (like his casualties of his first day, plus all the rearguards he’d need to disengage and withdraw).
Also bear in mind that Days 2 and 3, while Southern defeats, threw the Northern army into such disarray that it was incapable of chasing them when they finally retreated. It’s not so certain that they would have been able to disengage after Day 1 even if they had wanted to.
Remember, Lee believed in attack, and had made his reputation pressing Union generals aggressively. From Lee’s perspective, the first day had been a victory, although limited, and by following through against presumably off-balance and panicky foes, there was a chance to turn it into a decisive battle.