Hi Heraldeer 2
From what you have said, it doesn't sound like they've actually got to the disciplinary stage yet. A suspension frequently comes before the disciplinary process allowing investigations to be carried out. Neither does a suspension doesn't indicate guilt or that you will be dismissed. They don't have to give you a warning to suspend you.
You will likely receive an invitation to attend a discplinary hearing, have your say etc before a decision is reached on your future and any warning issued.
You need to check your contract also - have they suspended you on full pay? They should have...if not or if they don't proceed as outlined then you have a complaint.
As suggested, in preparation for a disciplinary hearing check your contract and the computer use policy.
In answer to whether it constitutes gross misconduct, I've seen it - at the end of the day you're paid to work not to spend all day on the internet.
Hope it works out