Plotly is always going to put the traces on top of the grid lines, so it doesn't matter what you do with the grid lines when the entire plot is filled in.

Alternatively, you can use shapes to mimic grid lines.

Here is an example with simulated gridlines for the y-axis.

hlines <- function(y = 0, color = "white") {
  list(type = "line",
       x0 = 0, x1 = 1, xref = "paper", y0 = y, y1 = y, 
       line = list(color = color, width = 1))
}
fig$x$layout$shapes <- lapply(-10:10, hlines)
fig

Thank you all for your help. Much appreciated. Below I document the full answer.

contour plot example with cartesian grid lines

rm(list=ls(all=TRUE))
library(plotly)
x1 <- -10:10
y1 <- -10:10
z1 <- sqrt(outer(x1 ^ 2, y1 ^ 2, "+"))
fig <- plot_ly(z = z1, x = x1, y = y1,  type = "contour", 
               line = list(width = 1.5, color = "black"),
               contours = list(showlabels = TRUE))
fig

# adding cartesian horizontal grid lines

hlines <- function(y = 0, color = "white") {
  list(type = "line",
       x0 = 0, x1 = 1, xref = "paper", y0 = y, y1 = y, 
       line = list(color = color, width = 0.5))
}
fig$x$layout$shapes <- lapply(-10:10, hlines)
fig


#Add vertical lines
vlines <- function(x = 0, color = "white") {
  list(type = "line",
       y0 = 0, y1 = 1, yref = "paper", x0 = x, x1 = x,
       line = list(color = color, width = 0.5))
}

fig$x$layout$shapes[22:42] <- lapply(-10:10, vlines)
fig