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I am trying to find a natural way to define $\pmod{1}$ over $\mathbb{R}$, and I would do this the same way as I would over the integers (with $\pmod{n}$ is defined as taking the quotient group $\mathbb{Z}/(n)$, where $(n)$ is the ideal generated by $n$), but $(1)\neq\mathbb{Z}$ so I can't write $\mathbb{R}/(1)$.

Is there a way to unify these definitions?

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    $\mathbb R/\mathbb Z$ is a well-defined additive abelian group. Is that what you want? – 2017-02-02
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    Maybe $\mathbb R/\mathbb Z$? – 2017-02-02

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You've got the right idea, $\Bbb Z/(n)$ is $\Bbb Z$ modulo the subgroup generated by $n$, so the same is true for $\Bbb R\mod 1$, it's $\Bbb R/(1)=\Bbb R/\Bbb Z$ so two real numbers are equivalent $\mod 1$ iff $x-y\in\Bbb Z$.

Note here this is as a group not as an ideal, I think that's where you're stumbling. This is inherently a group operation, not one in rings. And in particular this is a good thing, because $\Bbb R$ is a field so if this were a question about an ideal to make a quotient ring you would be SOL because the only ideals of fields, $F$, are $\{0\}$ and $F$ so any proper quotient would be trivial and therefore useless for your purposes.