I have already solved the problem.
Having looked at the *.so files (PHP extensions) in /usr/lib64/php/modules/, I noticed that access rights for them all except imagick.so were 755, and imagick.so had 644. That is, nobody was allowed to run this file.
I changed the access rights for imagick.so to 755, then restarted httpd and php:
sudo service httpd restart
sudo service php-fpm restart
After that, imagick started working and appeared in phpinfo() extensions list.