The incidents happened on the 1,360-passenger ferry Pride of Rotterdam on an overnight trip from Hull in England to Rotterdam, the world’s fifth-largest port.
The first incident happened around midnight (2300 GMT) when a 25-year-old Briton allegedly snatched a slice of pizza from another, sparking an argument.
“This led to a scuffle in which the 25-year-old man fell down a flight of stairs, seriously injuring his neck in the process,” police said in a statement.
Two Britons aged 23 and 27 were arrested in connection with the fight upon arrival in Rotterdam.
In a second incident at around 3am (0200 GMT) Sunday, four British men got into a fight in which an 18-year-old teen was stabbed in the neck with a piece of glass.
A 26-year-old man and two other 18-year-olds were arrested on arrival in port, police said.
“Both victims were seriously injured and airlifted by helicopter to a local hospital,” Rotterdam police spokesman Loek Houtepen told AFP.