EDIT: Removed all code and references to UIButton.
Thanks to @Matteo_Pacini for his answer to this question for showing us this technique. As with his answer (and comment), (1) this is rough around the edges and (2) I'm not sure this is how Apple wants us to use UIViewControllerRepresentable and I really hope they provide a better SwiftUI ("SwiftierUI"?) replacement in a future beta.
I put in a lot of work in UIKit because I want this to look good on an iPad, where a sourceView is needed for the popover. The real trick is to display a (SwiftUI) View that gets the UIActivityViewController in the view hierarchy and trigger present from UIKit.
My needs were to present a single image to share, so things are targeted in that direction. Let's say you have an image, stored as a @State variable - in my example the image is called vermont.jpg and yes, things are hard-coded for that.
First, create a UIKit class of type `UIViewController to present the share popover:
class ActivityViewController : UIViewController {
    var uiImage:UIImage!
    @objc func shareImage() {
        let vc = UIActivityViewController(activityItems: [uiImage!], applicationActivities: [])
        vc.excludedActivityTypes =  [
            UIActivity.ActivityType.postToWeibo,
            UIActivity.ActivityType.assignToContact,
            UIActivity.ActivityType.addToReadingList,
            UIActivity.ActivityType.postToVimeo,
            UIActivity.ActivityType.postToTencentWeibo
        ]
        present(vc,
                animated: true,
                completion: nil)
        vc.popoverPresentationController?.sourceView = self.view
    }
}
The main things are;
- You need a "wrapper" UIViewControllerto be able topresentthings.
- You need var uiImage:UIImage!to set theactivityItems.
Next up, wrap this into a UIViewControllerRepresentable:
struct SwiftUIActivityViewController : UIViewControllerRepresentable {
    let activityViewController = ActivityViewController()
    func makeUIViewController(context: Context) -> ActivityViewController {
        activityViewController
    }
    func updateUIViewController(_ uiViewController: ActivityViewController, context: Context) {
        //
    }
    func shareImage(uiImage: UIImage) {
        activityViewController.uiImage = uiImage
        activityViewController.shareImage()
    }
}
The only two things of note are:
- Instantiating ActivityViewControllerto return it up toContentView
- Creating shareImage(uiImage:UIImage) to call it.
Finally, you have ContentView:
struct ContentView : View {
    let activityViewController = SwiftUIActivityViewController()
    @State var uiImage = UIImage(named: "vermont.jpg")
    var body: some View {
        VStack {
            Button(action: {
                self.activityViewController.shareImage(uiImage: self.uiImage!)
            }) {
                ZStack {
                    Image(systemName:"square.and.arrow.up").renderingMode(.original).font(Font.title.weight(.regular))
                    activityViewController
                }
        }.frame(width: 60, height: 60).border(Color.black, width: 2, cornerRadius: 2)
            Divider()
            Image(uiImage: uiImage!)
        }
    }
}
Note that there's some hard-coding and (ugh) force-unwrapping of uiImage, along with an unnecessary use of @State. These are there because I plan to use `UIImagePickerController next to tie this all together.
The things of note here:
- Instantiating SwiftUIActivityViewController, and usingshareImageas the Button action.
- Using it to also be button display. Don't forget, even a UIViewControllerRepresentableis really just considered a SwiftUIView!
Change the name of the image to one you have in your project, and this should work. You'll get a centered 60x60 button with the image below it.