Meniscus-confined electrodeposition (MCED) is a simple and economical fabrication method for micro/nanoscale three-dimensional metal printing. In most applications of MCED of copper pillars using a deposition pipette probe, there exists a certain probe retraction speed range that can ensure successful pillar deposition. If the probe retraction speed exceeds this range, however, the morphological changes in the contact meniscus droplet bridge between the probe tip and substrate induce deposition current fluctuations. These fluctuations result in uneven pillar diameters (i.e., beaded wire deposition) and can even lead to nanowire breakage and growth termination. To analyze the MCED process, therefore, this paper first proposes a circuit model for the MCED process and then, based on this model, analyzes the mechanism underlying the deposition current fluctuations present as the probe retracts at higher speeds. To effectively suppress these deposition current fluctuations and improve the stability and quality of deposited copper pillars, a closed-loop constant-deposition-current control method is proposed. Compared with deposition using no control strategy, the constant-deposition-current control method is shown by simulations and experiments to significantly suppress the fluctuation of the deposition current and increase the stability of copper pillar growth. In addition, the constant-deposition-current control method is used for the rapid fabrication of copper pillars with high aspect ratios.