It could, but it was unlikely.

If we were to simply compare the armies they seem evenly matched. Hard numbers can be a bit tricky because both armies fielded troops of various qualities, for example the Garde Nationale in France, or the forces of the German allies who weren't as efficient as the prussians, so it's not exactly a 1 to 1 ratio when comparing the two. However once fully mobilised at the start of the conflict France had 900 000 men while Prussia and it's allies had 1 200 000. The training of the soldiers and their experience was also largely equivalent as both countries had fought several major wars prior to this conflict.

In terms of equipement both belligerants had their strength and weaknesses, Prussia had the upper hand in the artillery. Their cannons were both more numerous, precise and had a longer ranger. However France had far better rifles and also had machine guns, which Prussia completly lacked. Another strength of the French Empire was it's navy, which completly dwarfed it's Prussian counterpart and made a maritime blockade a real threat.

The main elements that gave Prussia a decisive edge were : logistics and commandement.

The logisitcs of the French army were a complete mess, the mobilization suffered a lot from this. Although both army had comparable numbers France could only muster at first 300 000 men against 500 000 Prussians plus their German allies. The main culprit are France railroad network which weren't as extensive as their German counter part and their antiquated mobilisation system.

The other main Prussian strength was commadement. The French generals, appart maybe from Mac Mahon, are completly outclassed by their Prussian counterparts. They still largely rely on Napoleon era tactics most importantly the division of forces into various corps d'armée lead by a largely independant general while the Prussian use a modern general-staff with Moltke firmly at the helm, allowing for better coordinated movement while the French fall victim to incoherent moves and strategies, most famously Bazaine decision to surrender in Metz in October.

With all that being said, could France have won the war of 1870 ? I think so, the French army was still among the greatest armies of the time and probably the closest match to the Prussians. It had it's advantages, most notably in it's infantry whose weapons largely outclassed their Prussian equivalents, and was able to score several localised victories. However the Prussian advantages in logistics and leadership were far more important in the strategical aspect, while France owns strength could only at best secure tactical victories. As such a French victory although possible, war is a chaotic affair, was unlikely. It would have most likely come down to a massive mistake by the Prussians at some point, or a massive hit of bad luck for them.

What would have Napoleon III done if he had won ? It's hard to tell. He didn't want that war and was pushed into it by Bismarck machinations and his republican opponents in parlement. He was also technically the agressors, and a French emperor, as such he would probably not get away with a peace treaty as harsh as the treaty of Frankfurt. He would most likely have used this victory to consolidate his regime internally and drove a wedge between the Northern and Southern German states. Maybe he would have take some territories in the Rhineland, like the Saarland, but not all of it. Another unforseen consequence could be an annexation of Luxembourg at a later date, as Napoleon III tried to do it before the war but was stopped by Prussia.