Actually, I managed to make it work, but you need okHttp library for that.
Try this when you're setting up browser activity:
    WebViewClient client = new WebViewClient() {
        private OkHttpClient okHttp = new OkHttpClient.Builder().build();
        @Override
        public WebResourceResponse shouldInterceptRequest(WebView view, String url) {
            Request okHttpRequest = new Request.Builder().url(url).build();
            try {
                Response response = okHttp.newCall(okHttpRequest).execute();
                return new WebResourceResponse(response.header("Content-Type", "plain/text"), response.header("Content-Encoding", "deflate"), response.body().byteStream());
            } catch (IOException e) {
                e.printStackTrace();
            }
            return null;
        }
    };
    webView.setWebViewClient(client);
Also, you'll need classic Trust Manager Manipulator, SSL socket factory and its implementation in your Application class:
public class TrustManagerManipulator implements X509TrustManager {
    private static TrustManager[] trustManagers;
    private static final X509Certificate[] acceptedIssuers = new X509Certificate[] {};
    public boolean isClientTrusted(X509Certificate[] chain) {
        return true;
    }
    public boolean isServerTrusted(X509Certificate[] chain) {
        return true;
    }
    public static void allowAllSSL()
    {
        HttpsURLConnection.setDefaultHostnameVerifier(new HostnameVerifier() {
            public boolean verify(String hostname, SSLSession session) {
                return true;
            }
        });
        SSLContext context = null;
        if (trustManagers == null) {
            trustManagers = new TrustManager[] { new TrustManagerManipulator() };
        }
        try {
            context = SSLContext.getInstance("TLS");
            context.init(null, trustManagers, new SecureRandom());
        } catch (NoSuchAlgorithmException e) {
            e.printStackTrace();
        } catch (KeyManagementException e) {
            e.printStackTrace();
        }
        HttpsURLConnection.setDefaultSSLSocketFactory(context
                .getSocketFactory());
    }
    public void checkClientTrusted(X509Certificate[] chain, String authType)
            throws CertificateException {
    }
    public void checkServerTrusted(X509Certificate[] chain, String authType)
            throws CertificateException {
    }
    public X509Certificate[] getAcceptedIssuers() {
        return acceptedIssuers;
    }
}
SSl Socket Factory:
public class TLSSocketFactory extends SSLSocketFactory {
    private SSLSocketFactory internalSSLSocketFactory;
    public TLSSocketFactory() throws KeyManagementException, NoSuchAlgorithmException {
        SSLContext context = SSLContext.getInstance("TLS");
        TrustManager[] managers = new TrustManager[] { new TrustManagerManipulator() };
        context.init(null, managers, new SecureRandom());
        internalSSLSocketFactory = context.getSocketFactory();
    }
    @Override
    public String[] getDefaultCipherSuites() {
        return internalSSLSocketFactory.getDefaultCipherSuites();
    }
    @Override
    public String[] getSupportedCipherSuites() {
        return internalSSLSocketFactory.getSupportedCipherSuites();
    }
    @Override
    public Socket createSocket(Socket s, String host, int port, boolean autoClose) throws IOException {
        return enableTLSOnSocket(internalSSLSocketFactory.createSocket(s, host, port, autoClose));
    }
    @Override
    public Socket createSocket(String host, int port) throws IOException, UnknownHostException {
        return enableTLSOnSocket(internalSSLSocketFactory.createSocket(host, port));
    }
    @Override
    public Socket createSocket(String host, int port, InetAddress localHost, int localPort) throws IOException, UnknownHostException {
        return enableTLSOnSocket(internalSSLSocketFactory.createSocket(host, port, localHost, localPort));
    }
    @Override
    public Socket createSocket(InetAddress host, int port) throws IOException {
        return enableTLSOnSocket(internalSSLSocketFactory.createSocket(host, port));
    }
    @Override
    public Socket createSocket(InetAddress address, int port, InetAddress localAddress, int localPort) throws IOException {
        return enableTLSOnSocket(internalSSLSocketFactory.createSocket(address, port, localAddress, localPort));
    }
    private Socket enableTLSOnSocket(Socket socket) {
        if(socket != null && (socket instanceof SSLSocket)) {
            ((SSLSocket)socket).setEnabledProtocols(new String[] {"TLSv1.1", "TLSv1.2"});
        }
        return socket;
    }
}
App class:
public class App extends Application {
    private static App appInstance;
    @Override
    public void onCreate() {
        super.onCreate();
        setupSSLconnections();
    }
    private void setupSSLconnections() {
        try {
            HttpsURLConnection.setDefaultSSLSocketFactory(new TLSSocketFactory());
        } catch (KeyManagementException | NoSuchAlgorithmException e) {
            e.printStackTrace();
        }
    }
}