The US has not blamed Cuba for the alleged attacks, and the Cuban government has previously denied targeting embassy staff.
Meanwhile, the US expelled 15 Cuban diplomats, saying that Havana had failed to protect its employees, but Cuba said the move was “unjustified”.
The US government also suspended visa processing in Cuba indefinitely.
Speaking in Washington at a meeting of Cubans living in the US, Bruno Rodriguez said the allegations have caused a “serious deterioration in the relationship between both governments and both countries”.
“It’s unacceptable and immoral, from the point of view of the Cuban government, for people to be harmed by a difference between governments,” the minister added.
The reported health problems ranged from mild brain trauma and deafness to dizziness and nausea.
Secretary of State Rex Tillerson said in a statement on October 3: “The decision was made due to Cuba’s failure to take appropriate steps to protect our diplomats in accordance with its obligations under the Vienna Convention. This order will ensure equity in our respective diplomatic operations.”
At least 21 people working at the US’s embassy in Havana have reported health problems, ranging from mild brain trauma and deafness to dizziness and nausea.
Earlier reports suggested sonic attacks were to blame, but nothing has been proven.
Havana denies targeting embassy staff, and the US has not blamed the Cuban government for the suspected attacks.
Rex Tillerson added in his latest statement: “We continue to maintain diplomatic relations with Cuba, and will continue to co-operate with Cuba as we pursue the investigation into these attacks.”
Fidel Castro has published an open letter to Cubans in which he makes no mention of the historic reopening of the US embassy in Havana.
The former Cuban leader instead criticizes American foreign and economic policies since World War Two and accuses the US of owing Cuba millions of dollars.
The letter was published to mark Fidel Castro’s 89th birthday.
The US embassy will be reopened in Havana on August 14, with Secretary of State John Kerry attending.
Fidel Castro said the US owed Cuba money because of the trade embargo the US imposed on the communist-run island in 1960.
Cuba says the embargo – which it calls a blockade – is hugely damaging to its economy.
The letter says relations will only be fully restored once it is lifted.
Three marines who lowered the American flag for the last time on January 4, 1961, will raise it again during Friday’s ceremony in Havana.
They are now retired and in their late 70s.
“I’m gonna love seeing that flag go back up,” said former marine Jim Tracy, 78, on a State Department video.
Cuba reopened its embassy in Washington last month.
In his birthday letter published in state newspaper Granma, Fidel Castro says Cuba is committed to “good will and peace in our hemisphere” but adds: “We will never stop fighting for the peace and welfare of all human beings, regardless of the color of their skin and which country they come from.”
Fidel Castro led his country from the Cuban Revolution, in 1959, until 2006, when he stood down because of undisclosed health problems.
He passed on power to his younger brother, Raul Castro, who embarked on a number of economic reforms.
After Raul Castro and President Barack Obama announced in December that Cuba and the US had agreed to restore diplomatic relations, it took Fidel Castro more than a month to express lukewarm approval for the historic reconciliation.