"acquiesced with" is correct and usable in written English.
You can use it when you want to say that someone agreed to something or went along with something. Example sentence: "The board members acquiesced with the decision to include a clause in the new contract."
The Franciscans acquiesced, with the stipulation that she sign a confidentiality agreement.
Mr. Harvey acquiesced, with one condition: "O.K., but I'm on the guest list when you play Madison Square Garden".
To appease them, therefore, the five acquiesced with the creation of Offa - providing that Offa was toothless.
In 2010 the controversial whistleblowing site Wikileaks released numerous documents relating to Pakistan which showed the Pakistani military and other arms of the government had "quietly acquiesced" with drone strikes even though they had publicly condemned them.
Pakistan's army and other arms of the government "quietly acquiesced" with drone strikes in the north of the country, the documents state, while condemning them in public as a breach of sovereignty.
The availability of the paper was controlled by Atlantic Productions, the organizers of the media frenzy, and despite the paper being available the day before the press conference, the management of PLoS One acquiesced with the request of the paper's authors that the research not be distributed to journalists under an embargo (Zimmer 2009f).
Pakistan's army is not the only arm of government quietly acquiescing with the programme, however.
Google is trying to ensure it cannot be accused of inconsistency – grandstanding with China while acquiescing with others.
"In short, it asks people to acquiesce with the status quo, and that probably helps," it says.
"It was here," Gibbon says somewhat ambiguously, "that I suspended my religious enquiries, acquiescing with implicit belief in the tenets and mysteries which are adopted by the general consent of Catholics and Protestants".
Agnès — who, as a woman, cannot read — nervously complains that it is in prose, with no illustrations, as Boy calmly announces his work is done; all the while the Protector, silently reading of his betrayal by both his wife and the Boy, disintegrates emotionally, later acquiescing, with trembling voice, in her plea that he read the page aloud.
