Hide inventory message on product page based on available stock quantity

Yep! You can check using the get_stock_quantity() method.

function my_wc_hide_in_stock_message( $html, $text, $product ) {
    $availability = $product->get_availability();
    $stock_qty = $product->get_stock_quantity();

    if ( isset( $availability['class'] ) && 'in-stock' === $availability['class'] && $stock_qty > 1000 ) {
        return '';
    }

    return $html;
}

add_filter( 'woocommerce_stock_html', 'my_wc_hide_in_stock_message', 10, 3 );