Problems
- You need to use an if/elsestatement to check if rows were returned from the query
- If rows were returned then output the row
- If not then output notification to that effect
 
- You should be using prepared statements to protect your database from malicious user input
- You don't need to use a while loop to access the returned results
- Unless you're expecting multiple rows returned -- which I doubt
 
Solution(s)
Notes
I've adjusted your use of ternary logic and incorporated the NULL coalescent operator instead:
$search = $_POST["search"] ?? $_GET["search"]  ?? NULL;
// if     $_POST["search"] has data then set $search = $_POST["search"]
// elseif $_GET["search"]  has data then set $search = $_GET["search"]
// else                                  set $search = NULL
mysqli
// Set database variables
$db_host = "localhost";
$db_user = "root";
$db_pass = "some+password";
$db_name = "sprawdzarka";
// Enable error reporting
mysqli_report(MYSQLI_REPORT_ERROR | MYSQLI_REPORT_STRICT);
// Connect to the database
$mysqli = new mysqli($db_host, $db_user, $db_pass, $db_name);
// Set the search parameter to value from $_POST, $_GET, or NULL
$search = ( $_POST["search"] ?? NULL ) ?: ( $_GET["search"]  ?? NULL );
// Check to see if the search URL was submitted
if ( $search ){
    $sql   = "SELECT * FROM domeny WHERE domena = ?";  // Prepare query string
    $query = $mysqli->prepare($sql);                   // Prep the query
    $query->bind_param("s", $search);                  // Bind the URL
    $query->execute();                                 // Execute the query
    $query->store_result();                            // Store the returned records
    $query->bind_result($domena, $handlowiec);         // Bind returned columns to variables
    // Check to see if a result was returned
    if( $query->fetch() ){
        // Print out the table row
        echo "
            <tr>
                <td>Domain:     {$domena}    </td>
                <td>Belongs to: {$handlowiec}</td>
            </tr>
        ";
    }
    else{
        // Print out if no results were returned
        echo "This domain doesn't exist: <a href=\"{$search}\">{$search}</a>";
    }
}
PDO
// Set database variables
$db_host = "localhost";
$db_user = "root";
$db_pass = "some+password";
$db_name = "sprawdzarka";
// Connect to the database
$pdo = new pdo(
    "mysql:host={$db_host};dbname={$db_name}",
    $db_user,
    $db_pass,
    [
        PDO::ATTR_ERRMODE => PDO::ERRMODE_EXCEPTION,
        PDO::ATTR_EMULATE_PREPARES => FALSE
    ]
);
// Set the search parameter to value from $_POST, $_GET, or NULL
$search = ( $_POST["search"] ?? NULL ) ?: ( $_GET["search"]  ?? NULL );
// Check to see if the search URL was submitted
if ( $search ){
    $sql   = "SELECT * FROM domeny WHERE domena = ?";  // Prepare query string
    $query = $pdo->prepare($sql);                      // Prepare query
    $query->execute([$search]);                        // Execute query and bind parameters
    // Check to see if a result was returned
    if( $row = $query->fetchObject() ){
        // Print out the table row
        echo "
            <tr>
                <td>Domain:     {$row->domena}    </td>
                <td>Belongs to: {$row->handlowiec}</td>
            </tr>
        ";
    }
    else{
        // Print out if no results were returned
        echo "This domain doesn't exist: <a href=\"{$search}\">{$search}</a>";
    }
}