On further research, it seems likely that in the US, "declare a riot" is an informal language referring to what might better be described as a public safety order to disperse (see example below.) Another term sometimes used in legal contexts is a lawful discretionary order, though the later term is somewhat broader.
Courts in the US have generally recognized the power of police (and other public safety officers) to issue such orders under the broad discretionary authority granted them. The exercise of discretion in issuing such orders can be challenged after-the-fact, in court, and such challenges are generally judged on a case-by-case basis. While failure to comply with any lawful order (including an order to disperse) is generally a misdemeanor, resisting force used to enforce it can lead to felony charges such as resisting arrest and assault on an officer.
The phrases "declaring a riot" and "reading the riot act" seem to be tied to a (now obsolete) UK statute known as the Riot Act of 1715, which was effect until 1967. Because this was a statute, rather than Common Law, it never applied in the United States. The riot act was a particularly severe statute as it made failure to comply with the order a felony punishable by death (according to Wikipedia.) Reading of the exact text specified in the Riot Act served as required warning of the consequences of non-compliance.
Some US states or locals have statutes similar in intent, but not severity, to the Riot Act, but few if any require a specific text as the Riot Act did. Seattle WA (for example) has published local policy requiring a specific processes be used for crowd management, including a script for an announcement:
“I am (rank and name) of the Seattle Police Department. I am now issuing a public safety order to disperse and I command all those assembled at (specific location) to immediately disperse, which means leave this area. If you do not do so, you may be arrested or subject to other police action. Other police action could include the use of chemical agents or less-lethal munitions, which may inflict significant pain or result in serious injury. If you remain in the area just described, regardless of your purpose, you will be in violation of city and state law. The following routes of dispersal are available: (routes). You have (reasonable amount of time) minutes to disperse.”
Note: The Seattle PD consented to a court-supervised program to make its use-of-force processes more transparent. The above example probably represents good practice, but seems not to be a legal requirement, except perhaps for departments under similar court supervision.