In Beijing, China's Foreign Ministry said they had lodged "stern representations" with Washington about the new building and the visit of the senior US diplomat.
It was opened to conduct relations with Taiwan following Washington's decision to switch diplomatic recognition to Beijing.
The United States on Tuesday opened a new $256-million (€225-million) representative office in Taiwan's capital, in a move that is likely to increase tensions between the USA and China.
The site spreads over 6.5 hectares, including Chinese gardens, in Taipei's Neihu district.
Trump decided not to send cabinet-level officials to the opening ceremony to avoid further antagonising Beijing amid other tensions over trade and China's claims to much of the South China Sea, sources with ties to the USA and Chinese governments told the South China Morning Post earlier this month.
Royce is a wife of Republican Ed Royce, chairman of the House of Representatives Committee on Foreign Affairs, a politician who has promoted several pro-Taiwan bills, including the Taiwan Travel Act, South China Morning Post (SCMP) reports.
The American Institute in Taiwan, as the de facto U.S. embassy in Taipei is called, was officially declared open Tuesday morning, in a ceremony attended by senior USA diplomats and Taiwan President Tsai Ing-wen.
Before she departs on Thursday, Royce will also hold discussions with Taiwan government officials concerning partnerships and exchanges between Taiwan and the US said the AIT in a statement.
On Sunday, Heather Nauert, spokesperson for the U.S. Department of Statement, also sent her congratulations via twitter and said the new AIT complex is "a milestone that reflects the importance of the U.S. -Taiwan partnership".
The new complex, a major upgrade from the low-key military building the AIT had used for decades, will serve as the representative office later this summer, said AIT Director Kin Moy.
Since Tsai came to power two years ago, Beijing has cut contact with her government because Taipei refuses to acknowledge that Taiwan is part of "one China".
The new building best represents the progress in bilateral ties made over the past years, but it will not officially begin operations until early September, the AIT said.
The Global Times, a widely-read Chinese state-run newspaper, said China should warn Taiwan and the United States against any provocation.