  We report a precise measurement of the isotope shifts in the $4^2$S$_{1/2}
\rightarrow 3^2$D$_{3/2}$ electric quadrupole transition at 732~nm in $^{40 -
42,44,48}$Ca$^+$ via high-resolution laser spectroscopy of co-trapped ions,
finding measured shifts of 2,775,392,374.8(6.0), 5,347,679,835.1(5.9), and
10,003,129,115.1(5.7)\,Hz between $^{42,44,48}$Ca$^+$and $^{40}$Ca$^+$,
respectively. When combined with prior measurements on the $4^2$S$_{1/2}
\rightarrow 3^2$D$_{5/2}$ transition [Phys. Rev. A 100, 022514 (2019),
https://journals.aps.org/pra/abstract/10.1103/PhysRevA.100.022514] a King Plot
analysis shows the data to be consistent with linearity below the level of
parts per billion. This observed linearity, which is free of nuclear
systematics, improves the previous isotope-shift based limits of Ca$^+$ for
couplings of a scalar boson beyond the Standard Model to electrons and neutrons
by a factor of 3. Our new limit excludes part of the coupling range remaining
for a new physics interpretation after accounting for one higher-order nuclear
term in the nonlinear King plot of Yb/Yb$^+$.
