It said he received the arms from a Gaza man employed at a French cultural centre in Gaza and brought them to someone in the West Bank where they were then sold to arms dealers.
"This is a very serious incident in which the immunity and privileges granted to foreign diplomatic missions in Israel were cynically exploited to smuggle dozens of weapons that may be used for terrorist attacks against Israeli civilians and security forces", the statement said. "Not for me to judge, I leave that all up to you". "We will not accept that, and we will not let it pass".
In response to Abbas, Friedman, who is Jewish, told a conference that Abbas's comments could have anti-Semitic connotations.
U.S. Middle East envoy Jason Greenblatt issued a statement calling Abbas' comment "highly inappropriate". His response was to refer to me as son of a dog.
"Some say wait for their plan", Abbas said, regarding the as-yet-undisclosed Trump peace plan.
"Notwithstanding his highly inappropriate insults against members of the Trump administration, the latest iteration being his insult of my good friend and colleague Ambassador Friedman, we are committed to the Palestinian people and to the changes that must be implemented for peaceful coexistence", he went on.
The crossing between Israel and Gaza is permanently on high alert, and Gaza has been subject to tight restrictions on movement and trade since the militant Islamist group Hamas took control of the enclave in 2007. The US has been pushing for progress in reconciliation in the run-up to its peace proposal.
The comments came amid roiling Palestinian anger at the USA over the Jerusalem decision, with Ramallah rejecting Washington's role in attempting to broker peace talks.
Abbas, however, said he was furious over the bomb that targeted the convoy carrying his prime minister and security chief last week, which did not seriously injure anyone.
"In my capacity as the president of the Palestinian people I have tolerated much in order to regain unity and unite the homeland and I was met with rejection by Hamas and their illegitimate authority", he said.
Israel has occupied the West Bank and east Jerusalem since 1967 and Mr Abbas's government has limited autonomy in parts of the West Bank, while the Jewish state annexed east Jerusalem in a move never recognised by the worldwide community.
Abbas has previously taken a series of measures, including reducing electricity payments for Gaza's two million residents, in what analysts said was an attempt to punish Hamas. "After 10 years, they realized that the Gaza humanitarian situation is tough", Abbas said derisively.