Yes, this sounds like international parental child abduction.
Please do not accept the response that “he has parental responsibility too, therefore there is nothing anyone can do.” Parental responsibility does not give one parent the automatic right to permanently remove a child from the UK and relocate her overseas without the consent of the other parent with parental responsibility or permission from the court.
I have unfortunately had experience of my own child being withheld from me by her father, although mine remained in the UK, and my strongest advice is to stop trying to reason with him and start creating a very clear legal and evidential trail immediately.
Screenshot and save absolutely everything, particularly the message where he says he is at the airport, is moving to America and will not be bringing your daughter back. Save your call logs, messages, flight information if you know it, passport details, his US address/employer if known, details of any relatives there and anything showing where your daughter normally lives, goes to school and has her life here.
Contact a specialist family solicitor urgently and specifically tell them that this is an international child abduction / Hague Convention case involving a six-year-old child habitually resident in England who has been removed to the USA without your consent. Those exact facts matter. I would not wait for a normal family-law appointment several weeks away.
I would also contact Reunite International Child Abduction Centre and the UK Central Authority for international child abduction. The USA and UK operate the Hague Convention, which exists specifically to deal with children being wrongfully removed from or retained away from the country in which they normally live.
Go back to the police as well and ask for the matter to be recorded specifically as suspected international parental child abduction, rather than simply a disagreement between two parents. Having parental responsibility does not make an unauthorised permanent removal abroad automatically lawful. Ask for an incident/crime reference and, if necessary, ask for the matter to be reviewed by a supervisor.
If you already have a Child Arrangements Order, take a copy of that straight to your solicitor and the police. If the order says your daughter lives with you, or sets out her time with her father, that is obviously extremely important. Even without an order, however, you should still get specialist advice urgently.
I would also contact her school tomorrow morning and tell them exactly what has happened so there is a contemporaneous record that she has been removed unexpectedly and has not simply changed schools with your agreement.
Please try not to send him angry or threatening messages, however understandable your anger is. Keep anything you send short, calm and child-focused: that you do not consent to your daughter relocating to the USA, you require her immediate return, and you want confirmation that she is safe and where she is staying. Everything could potentially end up before a court.
And please act quickly. Hague Convention proceedings are specifically about returning a child to the country of their habitual residence so that longer-term parenting disputes can be decided there. This is not something I would leave for social services to resolve.
I’m so sorry you are going through this. I remember the absolute panic of not knowing whether my own daughter was going to be returned to me, and that was without her having been taken to another country. Get specialist legal help immediately and keep pushing this as an international child-abduction matter rather than allowing it to be dismissed as “both parents have PR”.