Schema.org microdata code check
Home page (TravelAgency)

Looks correct.

Note that you could (if you want) use this item on every page of your website, not only on the home page.

Typically, you could always use this item for marking up the header of each page. By using its url property, you always link back to the same URL (i.e., your home page’s URL), so consumers can understand that it’s always the same TravelAgency entity.

Product page (MedicalWebPage)

Your use of the properties specialty, aspect, name, description, and image is correct.

author property

The value of the author property needs to be the URL of the author, not the URL of the Schema.org type.

Schema.org expects an Organization/Person as value, so you could make this explicit by using:

&lt;span itemprop="author" itemscope itemtype="http://schema.org/TravelAgency"&gt;
  &lt;link itemprop="url" href="HOME PAGE URL" /&gt;
&lt;/span&gt;


If you don’t want to use so "much" markup, you could also simply link directly:

&lt;link itemprop="author" href="HOME PAGE URL" /&gt;


However, most sites have a link to the home page anyway (i.e., in the logo or the navigation), so you could reuse this link instead of duplicating it with link.

url property

The value of the url property would have to be the URL of this MedicalWebPage, not the URL of your homepage. 

It doesn’t hurt to have it, but it isn’t really needed in the first place (as it’s just the URL of the current document anyway).

You could, if you want, combine this with a possible rel-canonical in your head.

offers property

Using the offers property on MedicalWebPage would mean that you are offering this web page (e.g., for sale), as its definition says (bold emphasis mine):


  An offer to provide this item […]


This is probably not what you want to say, or is it?

Offer

itemOffered property

The itemOffered property expects a Product as value. This is not required (using text, like you do, is allowed), but if you want to go the extra mile, make it a Product item:

&lt;span itemprop="itemOffered" itemscope itemtype="http://schema.org/Product"&gt;
  &lt;span itemprop="name"&gt;NAME OF TREATMENT&lt;/span&gt;
&lt;/span&gt; 


span element can’t have a content attribute

In HTML5/Microdata, only the meta element can have a content attribute, so it’s not allowed on span.

Instead of &lt;span itemprop="priceCurrency" content="EUR"&gt;€&lt;/span&gt;, you might want to use:

&lt;meta itemprop="priceCurrency" content="EUR"&gt;€


And instead of &lt;span itemprop="price" content="600.00"&gt;600.00&lt;/span&gt;:

&lt;span itemprop="price"&gt;600.00&lt;/span&gt;