Water cannon and police on horseback were used to disperse about 1,000 people outside the Turkish consulate.
The minister was trying to win support among expatriates for a referendum on expanding Turkish presidential powers.
The Dutch government says such rallies would stoke tensions days before the Netherlands’ general election.
Fatma Betul Sayan Kaya, Turkey’s family minister, had arrived by road on Saturday ahead of her planned rally.
But she was denied entry to the consulate in Rotterdam.
Ms Kaya was then taken to the German border by police, Dutch Prime Minister Mark Rutte confirmed early on Sunday morning.
Some 1,000 protesters took to the streets outside the Turkish consulate in Rotterdam
Earlier, the Netherlands had barred Turkish Foreign Minister Mevlut Cavusoglu from entering the country by plane. He has now travelled to Metz in northern France to address a rally there on Sunday.
In a Facebook post, Mr Rutte said attempts to find a “reasonable solution” to the countries’ differences had proved “impossible”, while dismissing Ms Kaya’s arrival in Rotterdam as “irresponsible”.
Ms Kaya said on Twitter: “The world must take a stance in the name of democracy against this fascist act! This behaviour against a female minister can never be accepted.”
Turkish Prime Minister Binali Yildirim said on Sunday that Turkey would respond in the “harshest ways” to this “unacceptable behaviour”.
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Turkey’s Hurriyet newspaper later reported that Ms Kaya had flown from Cologne back to Istanbul.
Turkish Prime Minister Recep Tayyip Erdogan had earlier denounced the Dutch government as “Nazi remnants and fascists” for denying the rallies.
The BBC’s Turkey correspondent, Mark Lowen, says that what began as a diplomatic row has turned into a full-blown bilateral crisis.
What is the row about?
Turkey is holding a referendum on 16 April on whether to turn from a parliamentary to a presidential republic, more akin to the United States.
If successful, it would give sweeping new powers to the president, allowing them to appoint ministers, prepare the budget, choose the majority of senior judges and enact certain laws by decree.
President Erdogan is hoping to win sweeping new powers
What’s more, the president alone would be able to announce a state of emergency and dismiss parliament.
In order to get it passed, Mr Erdogan needs the votes of citizens living within Turkey and abroad.
There are 5.5 million Turks living outside the country, with 1.4 million eligible voters in Germany alone – and the Yes campaign is keen to get them on side.
So a number of rallies have been planned for countries with large numbers of expat voters, including Germany, Austria and the Netherlands.
However, Mr Erdogan’s supporters have found themselves blocked from holding these rallies.
Why are countries trying to prevent the rallies?
Many of the countries have cited security concerns as the official reason.
Austrian Foreign Minister Sebastian Kurz said Mr Erdogan was not welcome to hold rallies as this could increase friction and hinder integration.
Mr Rutte said the Netherlands asked Turkey to desist as they feared “compromised public order and security”.
The Dutch government is also facing a severe electoral challenge from the anti-Islam party of Geert Wilders in its election on Wednesday.
Many European nations have also expressed deep disquiet about Turkey’s response to the July coup attempt and the country’s perceived slide towards authoritarianism under President Erdogan.
Germany in particular has been critical of the mass arrests and purges that followed – with nearly 100,000 civil servants removed from their posts.