No, and independent Roma would not ease a budgetary burden on host nations, because Roma isn't that large, and compared to the GDP of a whole nation, it is not a large burden. Additionally, having Roma is a privilege. It is an honor to host Roma, and it is not something that should be given up for symbolic reasons.
Although I support the concept of a Roma nation-state, I believe such a country would be close to third-world levels of poverty and dysfunction in its initial phases, as poverty and lack of education are currently prevalent through Roma communities. Therefore such a country would most likely require massive subsidies from Europe, thus not reducing any budgetary burdens.