Does this achieve the behavior you need?

cormat |> 
  as_tibble() |> 
  mutate(Var1 = rownames(cormat)) |> 
  pivot_longer(names_to = "Var2", values_to = "val", mpg:qsec)

Output:

# A tibble: 36 x 3
   Var1  Var2    val
   <chr> <chr> <dbl>
 1 mpg   mpg    1   
 2 mpg   disp  -0.85
 3 mpg   hp    -0.78
 4 mpg   drat   0.68
 5 mpg   wt    -0.87
 6 mpg   qsec   0.42
 7 disp  mpg   -0.85
 8 disp  disp   1   
 9 disp  hp     0.79
10 disp  drat  -0.71
# ... with 26 more rows

The nearly automated outputs from corrplot::corrplot() might get you what you want with very litle effort. Check out the vingette for more info.

library(tidyverse)
library(corrplot)

mtcars[, c(1,3,4,5,6,7)] %>% 
  cor(.) %>% 
  round(2) %>% 
  corrplot(method = "color", 
           type = "upper", 
           addCoef.col = "black",
           col = colorRampPalette(c("red", "white", "blue"))(200),
           tl.col = 'black')

^{Created on 2022-01-12 by the reprex package (v2.0.1)}

Compare str(melted_cormat) with str(pivoted_cormat). You'll find that the older reshape2::melt() converts strings to factors whereas tidyr::pivot_longer() leaves them strings.

The consequence of this is that with the melted version, ggplot() will order the rows and columns according to the factor levels and thus preserve the original order in cormat, but in the second case where they're just plain strings, they just go in alphabetical order.

To fix this, simply mutate() Var1 and Var2 into factor using as levels the original order of the columns in cormat. This will give you the plot you wanted.

Observe the difference in the last two lines of the example below and also note that the default method of cor is "pearson" so just be careful when you label your legend with the correlation method.

# Load packages
library(tidyverse)
library(reshape2)

# define plotting function
plot_fun <- function(dat) {
  ggplot(data = dat, aes(Var2, Var1, fill = value)) +
    geom_tile(color = "white") +
    scale_fill_gradient2(
      low = "blue",
      high = "red",
      mid = "white",
      midpoint = 0,
      limit = c(-1, 1),
      #space = "Lab",
      name = "Spearman\nCorrelation"
    ) +
    theme_minimal() +
    coord_fixed() +
    geom_text(aes(Var2, Var1, label = value),
              color = "black",
              size = 4) +
    theme(
      axis.text.x = element_text(
        family = "Calibri",
        face = "plain",
        color = "black",
        size = 12,
        angle = 0
      ),
      axis.title.x = element_blank(),
      axis.title.y = element_blank(),
      panel.grid.major = element_blank(),
      panel.border = element_blank(),
      panel.background = element_blank(),
      axis.ticks = element_blank(),
      legend.justification = c(1, 0),
      legend.position = c(0.9, 0.3),
      legend.direction = "horizontal"
    ) +
    guides(fill = guide_colorbar(
      barwidth = 7,
      barheight = 1,
      title.position = "top",
      title.hjust = 0.5
    ))
}

# Get the correlation matrix
cormat <- mtcars[, c(1, 3, 4, 5, 6, 7)] %>%
  cor(., method = "spearman") %>% # note selection of correlation method
  round(2) %>%
  replace(upper.tri(.), NA)

# make melted version
melted <- cormat %>%
  melt(na.rm = TRUE)

# make pivoted version
pivoted <-
  cormat %>%
  as.data.frame() %>%
  rownames_to_column("Var1") %>%
  pivot_longer(
    -Var1,
    names_to = "Var2",
    values_to = "value",
    values_drop_na = TRUE
  )

# note column types on melted vs pivoted
str(melted)
#> 'data.frame':    21 obs. of  3 variables:
#>  $ Var1 : Factor w/ 6 levels "mpg","disp","hp",..: 1 2 3 4 5 6 2 3 4 5 ...
#>  $ Var2 : Factor w/ 6 levels "mpg","disp","hp",..: 1 1 1 1 1 1 2 2 2 2 ...
#>  $ value: num  1 -0.91 -0.89 0.65 -0.89 0.47 1 0.85 -0.68 0.9 ...
str(pivoted)
#> tibble [21 x 3] (S3: tbl_df/tbl/data.frame)
#>  $ Var1 : chr [1:21] "mpg" "disp" "disp" "hp" ...
#>  $ Var2 : chr [1:21] "mpg" "mpg" "disp" "mpg" ...
#>  $ value: num [1:21] 1 -0.91 1 -0.89 0.85 1 0.65 -0.68 -0.52 1 ...

# melted version gives desired plot
melted %>% 
  plot_fun()

# pivoted version orders variables in alphabetical order
pivoted %>% 
  plot_fun()

# turning the variable names into a factor fixes the plot
pivoted %>% 
  mutate(across(starts_with("Var"), ~factor(.x, levels = colnames(cormat)))) %>%
  plot_fun()

^{Created on 2022-01-12 by the reprex package (v2.0.1)}