Maybe my question is naive, but:
If my paper was publish as open access (for example in Springer - http://www.springer.com/gp/open-access), could I upload the paper in researchgate.net and similar sites?
Open access comes in two degrees: gratis open access, which is free online access, and libre open access, which is free online access plus some additional usage rights.These additional usage rights are often granted through the use of various specific Creative Commons licenses. Only libre open access is fully compliant with definitions of open access such as the Berlin Declaration on Open Access to Knowledge in the Sciences and Humanities.
See Gratis versus libre for more ideas.
As a result, it depends on the contract you agreed to when submitting your article to the open access journal.
Maybe. But not necessarily. It depends on the exact terms and conditions of the open access paper: there are many different variants (Springer use more than one, as do many other publishers). It also depends on the exact terms and conditions of the site you intend to upload to.
So there is no general answer, beyond this:
Read and understand the licence on the paper.
Read and understand the terms and conditions of the site you want to upload to.
Work out whether they're mutually consistent, and whether you want to accept the T&Cs of the site you want to upload to.
If you are uncertain, then with a truly open-access paper you can also choose to avoid the question by simply linking to the official publication page - there is often a direct PDF link, and putting that on an external webpage gives no significant difference in user experience than uploading the PDF directly.