British Ambassador Matthew Rycroft, whose country holds the presidency of the Security Council this month, told reporters "there was no consensus in the room" for the statement.
Diplomats said China and Russia had raised objections, but the diplomatic missions of the two countries did not respond to requests for information.
The proposed British-drafted statement "noted with concern renewed fighting in some parts of the country and stressed the importance of humanitarian access to all the areas."
We support the peace process and are one of the largest bilateral humanitarian aid donors, including in Rakhine state — Matthew Rycroft
Most persecuted minority
An official statement, reached by consensus, expressing concern could have led to further action, but the move by China and Russia — which both have veto power — was seen as a clear signal that Myanmar should be left off the council agenda.
UN rights officials have accused the Myanmar military of extrajudicial killings, gang rapes and probable ethnic cleansing during the campaign against the Rohingya.
The European Union called on Thursday for the UN to send an international fact-finding mission urgently to Myanmar to investigate allegations of torture, rapes and executions by the military against the Rohingya Muslims.
On Thursday a panel led by former UN chief Kofi Annan reported over 120,000 Rohingya have languished for years in squalid displacement camps, which he said Myanmar must close.
— Human Rights Watch (@hrw) March 15, 2017
The 1.1 million Rohingya are loathed by many from the Buddhist majority in Myanmar, who insist they are illegal immigrants from Bangladesh even though many have lived in the country for generations.
The UN has described the Rohingya as the most persecuted minority in the world.
Religious persecution in Myanmar has also targeted Christians. More than 100,000 Christians from Myanmar live in Malaysia as refugees after recently fleeing their homes.