Send notification to iPhone from Apple Watch in Swift

Yes.

I did it this way:

• create an IBAction (that I linked to a tap on a label).
• Here it is to ask for a sync between iPhone and Watch
• synced done is displayed on the Watch in:

    @IBOutlet weak var autonomyMessage: WKInterfaceLabel!   

IBAction:

    @IBAction func tapOnConnectionToIphone(_ sender: Any) {
        
        autonomyMessage.setText("")
        let session = WCSession.default
        if session.isReachable {
            session.sendMessage(["request" : "sync"], replyHandler: { (response) in
                self.autonomyMessage.setText(NSLocalizedString("Received!", comment: "SessionInterfaceController"))
            }, errorHandler: { (error) in
                self.autonomyMessage.setText(NSLocalizedString("Not received!", comment: "SessionInterfaceController"))
            })
            autonomyMessage.setText(NSLocalizedString("Sent!", comment: "SessionInterfaceController"))
        } else {
            autonomyMessage.setText(NSLocalizedString("Not sent!", comment: "SessionInterfaceController"))
        }
        autonomyMessage.setHidden(false)
    }

On the iPhone side:

    func session(_ session: WCSession, didReceiveMessage message: [String : Any], replyHandler: @escaping ([String : Any]) -> Void) {
        
        if message["request"] as? String == "sync" {
           // proceed as needed
       }
}

For SwiftUI, you'd only need to translate the Watch side of what was provided by @Claude31, similar to this:

struct ScheduleNotificationOnPhoneView: View {
    @State private var result = ""

    func sendMessageToPhone() {
        let session = WCSession.default
        
        guard session.isReachable else {
            result = "Session is not reachable."
            return
        }
        
        session.sendMessage(["request": "scheduleNotification"],
                            replyHandler: { _ in
            // You probably want to parse the reply here to handle the reply from your phone
            result = "Notification request sent to phone."
        }, errorHandler: {
            result = "Error sending request: \($0.localizedDescription)"
        })
    }
    
    var body: some View {
        VStack {
            Button("Schedule on Phone") {
                sendMessageToPhone()
            }
        }
    }
}

In UIKit case, iPhone side goes into the viewController that handles the message (in my case, the initial VC)