The concept of a day is quite ambiguous in any language and Japanese is no exception. A day can be: 24 hours, a calendar day, a period between sunrise and sunset, a period between you wake up and go to sleep, one's workday. One day might have more than 24 hours (長い一日だった), the other less...
That said 一日{いちにち} in your example refers to a time period that someone perceived or might perceive as a "beautiful day". It is essentially 一日間 here and although using it would be unnatural, it would probably be grammatical.
Like in 忙しい一日でした the 間 is absent, but if you extended the period to two weeks you would say 忙しい2週間でした.
日{ひ} on the other hand has more of a "calendar day" or an "exemplary day" feeling to it. うららかな日は外で遊びます。