Código iOS para salvar o arquivo “Abrir em” no diretório Documentos (e resolver nomes de arquivo duplicados)

Em primeiro lugar, você precisa configurar seu comportamento de “Abrir em” registrando o tipo de arquivo UTI. A partir daí siga o código comentado.

AppDelegate.m

- (BOOL)application:(UIApplication *)application openURL:(NSURL *)url sourceApplication:(NSString *)sourceApplication annotation:(id)annotation
{

    // Copy the file into the documents directory, renaming if there's a duplicate
    [self copyItemAtURLtoDocumentsDirectory:url];

    return YES;
}



-(BOOL)copyItemAtURLtoDocumentsDirectory:(NSURL*)url {

    NSError *error;
    NSURL *copyToURL =  [self applicationDocumentsDirectory];

    NSString *fileName = [url lastPathComponent];

    // Add requested file name to path
    copyToURL = [copyToURL URLByAppendingPathComponent:fileName isDirectory:NO];   

    if ([[NSFileManager defaultManager] fileExistsAtPath:copyToURL.path]) {

        // Duplicate path
        NSURL *duplicateURL = copyToURL;
        // Remove the filename extension
        copyToURL = [copyToURL URLByDeletingPathExtension];
        // Filename no extension
        NSString *fileNameWithoutExtension = [copyToURL lastPathComponent];
        // File extension
        NSString *fileExtension = [url pathExtension];

        int i=1;
        while ([[NSFileManager defaultManager] fileExistsAtPath:duplicateURL.path]) {

            // Delete the last path component
            copyToURL = [copyToURL URLByDeletingLastPathComponent];
            // Update URL with new name
            copyToURL=[copyToURL URLByAppendingPathComponent:[NSString stringWithFormat:@"%@–%i",fileNameWithoutExtension,i]];
            // Add back the file extension
            copyToURL =[copyToURL URLByAppendingPathExtension:fileExtension];
            // Copy path to duplicate
           duplicateURL = copyToURL;
            i++;

        }


    }

    BOOL ok = [[NSFileManager defaultManager] moveItemAtURL:url toURL:copyToURL error:&error];

    // Feed back any errors
    if (error) {
        NSLog(@"%@",[error localizedDescription]);
    }

    return ok;
}

-(NSURL *)applicationDocumentsDirectory {

    NSString *documentsDirectory;
NSArray *paths = NSSearchPathForDirectoriesInDomains(NSDocumentDirectory, NSUserDomainMask, YES);
if ([paths count] > 0) {
    documentsDirectory = [paths objectAtIndex:0];
}
// Important that we use fileURLWithPath and not URLWithString (see NSURL class reference, Apple Developer Site)
    return [NSURL fileURLWithPath:documentsDirectory];
}

Atualmente, isso coloca todo o código dentro do arquivo AppDelegate, o que, obviamente, provavelmente não seria sua escolha em um aplicativo funcional, mas pelo menos é uma introdução.