Can my referral buy my gig ?
Off course, but the referral links go to the Fiverr homepage, not directly to your Fiverr page. Think of it as a 2nd income stream apart from making sales from your own gigs, where you can earn money when someone registers on Fiverr and buys ANY gig available, not just yours.
It is likely technically possible BUT it does sound like gaming the marketplace and might be “rewarded” with account suspension.
As @blavaro has suggested, you might regret trying to do this.
As a longterm reader of the forums, I can recall threads from sellers saying they’ve had account warnings or been suspended for messaging themselves in order to try and improve their response rate. I really don’t think Fiverr will look too favourably on you trying to purchase your own gig, which is clearly an attempt to bypass Fiverr’s insistence that referral programme revenue can only be used to purchase gigs - and not directly withdrawn as earnings.
In fact, I’ve just found this from Fiverr’s referral programme page: Important Disclaimers
We reserve the right to suspend your account and remove referrals shopping credit entitlement should we notice any activity we determine contrary to the Referral Program terms and conditions or the Fiverr Terms of Service by the Referrer and/or Referral.
I’m not going to read through the whole page. You can do that - but I think you will come unstuck if you attempt your idea! Source: https://www.fiverr.com/referral_program