Specifically it’s from a speech on 5 October 1937 in Chicago. He begins talking about the devolving “political situation in the world.”
In this speech he often references how other countries trying to create war but I’m unfamiliar with these specific events in 1937 that FDR seems to believe are contributing to “world lawlessness.”
The two major wars in 1937 were the Spanish Civil War and the Second Sino-Japanese War.
Both were well covered in the contemporary news and both featured civilian massacres.
The Spanish Civil War had a "white terror" perpetrated by the Nationalist Spanish forces against the Basque, Communists, Anarchists and others.
The bombs from airplanes on civilians bit sounds like the Bombing of Guernica. Which was conducted by German/Nationalist aircraft against a Spanish Republican city in April 1937.
The Spanish Civil War was also a "red terror" perpetrated by Republican forces that targeted the Bourgeois, politicians, Priests, and others.
The Sino-Japanese War had many, many, massacres but the most well attested to is the Rape of Nanjing. Somewhere in the range of 50 to 300 thousand Chinese civilians were killed.
This is the most well documented atrocity during the Sino-japanese war due to the large western presence in the city.
The Rape of Nanjing began in December of 1937 so its several months after the speech. The war began in July and I am not sure if news of other Japanese atrocities had reached President Roosevelt by then.
Both wars also involved foreign powers giving supplies to armies of the two sides of the existing civil war.
To give one example: Nazi Germany supplied Nationalist Spain - German troops, supplies, and equipment against Republican Spain. This was to field test equipment, potentially gain an ally in Europe, and opposition to the leftist Republican forces.
Nazi Germany also supplied training and equipment to Nationalist China. This relationship was based on their mutual anti-communist ideologies, the communist Chinese uprising, and the immense resources a Chinese ally could offer. Ironically, these were primarily used against the Imperial Japanese Army who would become Germany's ally later.
To give a further single example: The 87th division was German trained and equipped and fought in the Battle of Shanghai against Imperial Japan in 1937. The 87th division also saw action against Communist China in the Chinese Civil War but that was not until much later.
This answer does not touch on the many other countries that supplied aid to the warring nations besides Nazi Germany.
Nor does it answer the naval atrocities portion of your question.
Both wars fit Roosevelt's vague phrasing but I would lean towards the Second Sino-Japanese war since it had just begun in 1937.
Here's an excellent comment that lists books on the Second Sino-Japanese war by /u/scipioasina