Use HTML Tidy to just indent HTML code?

You need the following option:

tidy --show-body-only yes -i 4 -w 80 -m file.html

http://tidy.sourceforge.net/docs/quickref.html#show-body-only

-i 4 - indents 4 spaces (EDIT: tidy never uses tabs)
or
--indent-with-tabs yes - instead (--tab-size may affect wrapping)

-w 80 - wrap at column 80 (default on my system: 68, very narrow)

-m - modify file inplace

(you may want to leave out the last option, and examine the output first)

Showing only body, will naturally leave out the tidy-mark (generator meta).

Another cool options are: --quiet yes - doesn't print W3C advertisements and other unnecessary output (errors still reported)

I didn't found a possibility "only reindent - without any changes". The next config file will "repair" as low as possible and (mostly) only re-indent the html. Tidy still correcting some errorish conditions, like duplicated (repeated) attributes.

#based on http://tidy.sourceforge.net/docs/quickref.html
#HTML, XHTML, XML Options Reference
anchor-as-name: no  #?
doctype: omit
drop-empty-paras: no
fix-backslash: no
fix-bad-comments: no
fix-uri:no
hide-endtags: yes   #?
#input-xml: yes     #?
join-styles: no
literal-attributes: yes
lower-literals: no
merge-divs: no
merge-spans: no
output-html: yes
preserve-entities: yes
quote-ampersand: no
quote-nbsp: no
show-body-only: auto

#Diagnostics Options Reference
show-errors: 0
show-warnings: 0

#Pretty Print Options Reference
break-before-br: yes
indent: yes
indent-attributes: no   #default
indent-spaces: 4
tab-size: 4
wrap: 132
wrap-asp: no
wrap-jste: no
wrap-php: no
wrap-sections: no

#Character Encoding Options Reference
char-encoding: utf8

#Miscellaneous Options Reference
force-output: yes
quiet: yes
tidy-mark: no

For example the next html-fragment

<div>
<div>
<p>
not closed para
<h1>
h1 head
</h1>
<ul>
<li>not closed li
<li>closed li</li>
</ul>
some text
</div>
</div>

will changed to

<div>
    <div>
        <p>
            not closed para
        <h1>
            h1 head
        </h1>
        <ul>
            <li>not closed li
            <li>closed li
            </ul>some text
    </div>
</div>

As you can notice, the hide-endtags: yes hides the closing </li> from the second bullet in the input. Setting the hide-endtags: no - will get the next:

<div>
    <div>
        <p>
            not closed para
        </p>
        <h1>
            h1 head
        </h1>
        <ul>
            <li>not closed li
            </li>
            <li>closed li
            </li>
        </ul>some text
    </div>
</div>

so, tidy adds closing </p> and closing </li> to first bullet.

I didn't found a possibility preserve everything on input and only reindent the file.

Use the indent, tidy-mark, and quiet options:

tidy \
  -indent \
  --indent-spaces 2 \
  -quiet \
  --tidy-mark no \
  index.html

Or, using a config file rather than command-line options:

indent: auto
indent-spaces: 2
quiet: yes
tidy-mark: no

Name it tidy_config.txt and save it the same directory as the .html file. Run it like this:

tidy -config tidy_config.txt index.html

For more customization, use the tidy man page to find other relevant options such as markup: no or force-output: yes.