If you want to prove the actual tax liability you have in the US - you have to file a tax return. If the Romanian government believes you that the withholding is your actual tax - fine, but that would be a lie. Withholding is not a tax.
The American payers must withhold from foreigners enough to have transferred more than the actual tax the foreigners would have paid. The standard withholding is 30%, but the actual tax on dividends varies. In case the tax treaty limits the tax on dividends - the withholding is usually up to the maximum of the tax allowed by the treaty. But allowed doesn't mean that would be the actual tax. In many cases it is not.
So if you want to claim the US tax paid as a credit towards your Romanian tax - you'll need to file a tax return in the US, calculate the actual tax liability and that would be the amount of credit you can claim. The difference between that amount and the amount withheld by the payer will be refunded to you by the IRS.
You don't have to file a US tax return, that is true. But the withholding is not the tax, the actual tax liability may have been less, and the Romanian tax authority may deny your credit, in whole or in part, based on the fact that you haven't filed a US tax return and as such have no proof of your actual tax paid.
You had some experience with the UK tax treaty, and you think all the treaties are the same. That may be a reasonable line of thought, but it is incorrect. Treaties are not the same. More importantly, even if the treaty is the same - the tax law is not. While in the UK the tax on dividends may be flat and from the first pound - in the US it is neither flat nor from the first dollar. Thus, while in the UK you may have been used to paying tax at source and that's it - in the US it doesn't work that way at all.