No. The selected date range does not affect whether a user is classified as new or returning.
A user is considered returning if they had their first session at any time in the past — even if that first session occurred outside of the selected date range.
For example, if your date range is December 1–15 and a user first visited on November 30 and then returned on December 3, that user would be counted as a returning visitor within the December date range.
In short, the date range only controls which sessions are included in the analysis, not how a user’s new vs. returning status is determined.