"I think that something wrong with file permissions."

I would agree. Help us, help you by providing your current permissions to the /wp-content/ directory.

How did you setup Wordpress? Do you get any errors when you try to import the file?

I setup WordPress following this guide -- https://www.digitalocean.com/community/articles/how-to-install-wordpress-with-nginx-on-centos-6--2. Than I have installed WordPress Importer plugin. Changed file permissions by running these commands -- chown -R root:root .
chown -R nginx:nginx wp-content wp-admin/update.php wp-admin/update-core.php
wp-admin/network/update.php wp-admin/network/update-core.php. When I'm trying to import, after choosing the file to upload and pressing 'Upload file and import' button, I'm getting white page, showing left vertical menu only. No error messages or something... What am I doing wrong?

Why did you chown to root:root and nginx:nginx? Shouldn't you chown -R www-data:www-data /.../wp-content?

Because I'm runnig CentOS and nginx. I thought that www-data is an Ubuntu default user.

@umfromthedarkside: What's the output of

sudo ps wwaux | grep fpm

?

root 1350 0.0 0.8 208280 4176 ? Ss 09:00 0:01 php-fpm: master process (/etc/php-fpm.conf)
nginx 1351 0.0 5.2 228068 26492 ? S 09:00 0:22 php-fpm: pool www
nginx 1352 0.0 5.2 228016 26448 ? S 09:00 0:17 php-fpm: pool www
nginx 1353 0.0 5.2 228068 26504 ? S 09:00 0:17 php-fpm: pool www
nginx 1354 0.0 5.1 227540 25824 ? S 09:00 0:20 php-fpm: pool www
nginx 1355 0.0 5.1 227536 25844 ? S 09:00 0:15 php-fpm: pool www
nginx 1422 0.0 4.8 226216 24512 ? S 09:01 0:22 php-fpm: pool www
nginx 1919 0.0 2.3 213668 11940 ? S 11:55 0:15 php-fpm: pool www
nginx 2043 0.0 4.9 226236 24652 ? S 12:47 0:12 php-fpm: pool www
nginx 2050 0.0 2.3 213660 11928 ? S 12:48 0:11 php-fpm: pool www
nginx 2051 0.0 4.8 226140 24436 ? S 12:48 0:12 php-fpm: pool www
nginx 2470 0.0 2.3 213680 11928 ? S 14:10 0:08 php-fpm: pool www
doxxxx 3072 0.0 0.1 6372 660 pts/0 S+ 18:36 0:00 grep fpm

Hmm. So php5-fpm is running as user 'nginx'. What's the output of

stat /path/to/wp-content

?