While this is slightly outside the scope of your question, Europeans did take advantage of the U.S. being divided before and during the war. As has been pointed out in other posts, the United States came out of the civil war with a strong economy and military.

A French, British and Spanish force invaded Mexico in December of 1861, originally to collect debts that the struggling Mexican government had refused to pay. The French government soon revealed that they hoped to seize the entire country and the their allies withdrew.

The French were initially successful. They were able to conquer much of the country, create a new monarchical Mexico government with the existing Mexican aristocracy and put the Austrian Emperor Maximilian I on the throne in 1864. This Catholic monarchy not only provided a counterbalance to the American Protestant republic, but provided the French access to Latin American markets and to Mexican silver. The Mexican aristocracy supported this plan because they believed that a monarch would bring stability to the country.

This French-supported Second Mexican Empire was not able to bring stability to the country. A rival republican Mexican government led by President Benito Juarez had survived in Veracruz. Although distracted by the Civil War, Congress refused to recognize Maximilian's government and passed a resolution condemning it before his coronation:

That the Congress of the United States are unwilling, by silence, to leave the nations of the world under the impression that they are indifferent spectators of the deplorable events now transpiring in the Republic of Mexico; and they therefore think fit to declare that it does not accord with the policy of the United States to acknowledge a monarchical government, erected on the ruins of any republican government in America, under the auspices of any European power."

As soon as the American Civil War ended, the American government began providing President Juarez with weapons and supplies. There were even suggestions of using an invasion of Mexico as a way to heal the rift with the former Confederate states. Facing down a United States, France soon withdrew support from Emperor Maximilian. The US-backed republican government's forces soon captured the emperor and he was executed.