Hungary, together with Poland, vetoed both the budget and the recovery plan on Monday, arguing that the budget law included a clause which makes access to money conditional on respecting the rule of law.
Orban is a staunch opponent of mass immigration on the grounds that it dilutes national and European identity. The EU has launched a procedure against Hungary for violating democratic norms.
"After accepting the current proposal there would be no barrier against tying the disbursement of funds to member states to supporting immigration and blackmailing those countries that oppose immigration with budgetary sanctions," Orban said in a statement published by state news agency MTI.
The budget and the recovery package do not have any specific clauses about immigration, and political analysts said Orban's comments were aimed at shoring up support among his domestic voters by using anti-immigration rhetoric again.
Orban said Hungary was committed to the rule of law, but added: "Those who defend their borders and defend their countries against immigration can no longer be classified as law-abiding in Brussels."
He added that Hungary had accepted the recovery fund and long-term budget package in July "because we are committed to European solidarity and support financial aid to the states that need it as quickly as possible".
The wording on conditionality in the political agreement accompanying the package was watered down on the insistence of Hungary and Poland.
Commenting on Orban's latest statement, Political Capital analyst Patrik Szicherle said: "This is a political narrative for a domestic audience.
"Since the migration crisis of 2015 Orban has traced any criticism of Hungary's government or (its record on) the rule of law back to a rejection of immigration or George Soros."
Orban's government has long demonised the Hungarian-born billionaire philanthropist, accusing him of encouraging mass immigration, something Soros denies. His support for an open liberal society clashes with Orban's nationalist ideology.
(Reporting by Marton Dunai; Editing by Gareth Jones)