For that, you can use Magento\Cms\Api\GetPageByIdentifierInterface. No need to use Magento\Framework\Api\SearchCriteriaBuilder.

use Magento\Cms\Api\GetPageByIdentifierInterface.

/**
* @var \Magento\Cms\Api\GetPageByIdentifierInterface
*/
private $pageByIdentifier;

public function __construct(
        GetPageByIdentifierInterface $pageByIdentifierInterface
    ) {
        $this->pageByIdentifier = $pageByIdentifier   
    }

Now you can get that page data passing pageIdentifier and storeId.

$page = $this->pageByIdentifier->execute($pageIdentifier, $storeId);