Sami Abu Zuhri had tried to cross into the Gaza Strip with the money but was detained and the cash confiscated, Julio de la Guardia, a spokesman for the European Union observation force at Rafah terminal, said on Friday.
Travellers reportedly cannot carry more than $2,000 without declaring it to the authorities.
Abu Zuhri initially refused to leave the airport, leading to a tense stand off between Hamas fighters who rushed to the scene and the Palestinian presidential guard at the airport.
Senior Abbas aide Saeb Erikat said the money had been sent to the Palestinian attorney general, who was asked by Abbas to investigate the incident.
But Hamas officials demanded that the money be returned, saying that Abu Zuhri was carrying money donated to the Hamas government and for Palestinians in Israeli jails.
"If bringing support for my people is a crime, then I am very proud of this crime," Abu Zuhri told reporters on Friday.
"We are upset to be dealt with this way at a time when the Palestinian people are suffering from siege and starvation."
Tensions between Abbas and the Hamas-led government have increased in recent days, after clashes on Thursday and Friday between Palestinian forces loyal to Abbas and Hamas' new security force.
Facing a US-led funding and banking blockade, the Hamas-led Palestinian Authority has been unable to pay salaries to government employees in recent months.
Hamas has been unable to receive funds donated by Arab governments because Arab banks, fearing repercussions from the US and other Western nations, will not process the transactions.
Western governments have refused to give direct aid to the Palestinian government until Hamas renounces the armed struggle and recognises Israel.